| L(s) = 1 | − 6·2-s + 4·3-s + 24·4-s + 7·5-s − 24·6-s + 7·7-s − 80·8-s − 11·9-s − 42·10-s + 74·11-s + 96·12-s + 107·13-s − 42·14-s + 28·15-s + 240·16-s + 24·17-s + 66·18-s + 228·19-s + 168·20-s + 28·21-s − 444·22-s − 149·23-s − 320·24-s − 117·25-s − 642·26-s − 187·27-s + 168·28-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 0.769·3-s + 3·4-s + 0.626·5-s − 1.63·6-s + 0.377·7-s − 3.53·8-s − 0.407·9-s − 1.32·10-s + 2.02·11-s + 2.30·12-s + 2.28·13-s − 0.801·14-s + 0.481·15-s + 15/4·16-s + 0.342·17-s + 0.864·18-s + 2.75·19-s + 1.87·20-s + 0.290·21-s − 4.30·22-s − 1.35·23-s − 2.72·24-s − 0.935·25-s − 4.84·26-s − 1.33·27-s + 1.13·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405224 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405224 ^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.641618758\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.641618758\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 + p T )^{3} \) |
| 37 | $C_1$ | \( ( 1 + p T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 - 4 T + p^{3} T^{2} + 35 T^{3} + p^{6} T^{4} - 4 p^{6} T^{5} + p^{9} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 - 7 T + 166 T^{2} - 392 p T^{3} + 166 p^{3} T^{4} - 7 p^{6} T^{5} + p^{9} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - p T + 491 T^{2} - 386 p T^{3} + 491 p^{3} T^{4} - p^{7} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 74 T + 3879 T^{2} - 136237 T^{3} + 3879 p^{3} T^{4} - 74 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 107 T + 7706 T^{2} - 405692 T^{3} + 7706 p^{3} T^{4} - 107 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 24 T + 10303 T^{2} - 297360 T^{3} + 10303 p^{3} T^{4} - 24 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 12 p T + 33277 T^{2} - 3116904 T^{3} + 33277 p^{3} T^{4} - 12 p^{7} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 149 T + 34472 T^{2} + 2934600 T^{3} + 34472 p^{3} T^{4} + 149 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 325 T + 103342 T^{2} + 16490600 T^{3} + 103342 p^{3} T^{4} + 325 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 23 T + 55040 T^{2} - 60886 T^{3} + 55040 p^{3} T^{4} - 23 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 464 T + 173745 T^{2} + 44599735 T^{3} + 173745 p^{3} T^{4} + 464 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 588 T + 188977 T^{2} - 50868960 T^{3} + 188977 p^{3} T^{4} - 588 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 155 T + 193607 T^{2} - 8850326 T^{3} + 193607 p^{3} T^{4} - 155 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 579 T + 502771 T^{2} + 172379378 T^{3} + 502771 p^{3} T^{4} + 579 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 1258 T + 708749 T^{2} - 308542444 T^{3} + 708749 p^{3} T^{4} - 1258 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 291 T + 240820 T^{2} - 157391290 T^{3} + 240820 p^{3} T^{4} - 291 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 127 T + 630358 T^{2} + 43389690 T^{3} + 630358 p^{3} T^{4} + 127 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 201 T + 392389 T^{2} + 290908338 T^{3} + 392389 p^{3} T^{4} + 201 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 42 T + 195091 T^{2} - 46732497 T^{3} + 195091 p^{3} T^{4} - 42 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 413 T + 798054 T^{2} - 284589394 T^{3} + 798054 p^{3} T^{4} - 413 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 1037 T + 1922285 T^{2} + 1185346766 T^{3} + 1922285 p^{3} T^{4} + 1037 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 660 T + 408043 T^{2} + 56148120 T^{3} + 408043 p^{3} T^{4} - 660 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 1384 T + 3043983 T^{2} - 2542109040 T^{3} + 3043983 p^{3} T^{4} - 1384 p^{6} T^{5} + p^{9} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.38629157726457219363435822230, −11.90054603684560371513123506662, −11.54833851774040129850558173696, −11.53792853550475425416381114684, −11.11424390203573125239288631243, −10.73398538248112268166131516710, −9.991396882815858988684437517264, −9.693623533096504050547234692913, −9.502657276706303082172623972497, −9.231115324466430228486916088747, −8.806655653872362639529637943363, −8.240512747768434046916387376937, −8.229113314591001882546121599070, −7.68307967775980104624470204821, −7.02332745766936583040110129147, −6.93133718285702884483625402325, −5.96372402165584654416789887286, −5.79983541067185468492168340028, −5.61424580293187930819073549423, −4.05221881292814315737281702146, −3.50179936631647083590577987510, −3.27368114020858683240359520402, −1.86114520505421590067682439614, −1.69156785543680390450576098167, −0.890230193606046397934341983531,
0.890230193606046397934341983531, 1.69156785543680390450576098167, 1.86114520505421590067682439614, 3.27368114020858683240359520402, 3.50179936631647083590577987510, 4.05221881292814315737281702146, 5.61424580293187930819073549423, 5.79983541067185468492168340028, 5.96372402165584654416789887286, 6.93133718285702884483625402325, 7.02332745766936583040110129147, 7.68307967775980104624470204821, 8.229113314591001882546121599070, 8.240512747768434046916387376937, 8.806655653872362639529637943363, 9.231115324466430228486916088747, 9.502657276706303082172623972497, 9.693623533096504050547234692913, 9.991396882815858988684437517264, 10.73398538248112268166131516710, 11.11424390203573125239288631243, 11.53792853550475425416381114684, 11.54833851774040129850558173696, 11.90054603684560371513123506662, 12.38629157726457219363435822230
