| L(s) = 1 | + 14·5-s − 17·9-s + 32·11-s + 28·13-s + 154·17-s + 224·19-s + 68·23-s + 45·25-s − 236·29-s + 196·31-s + 346·37-s + 420·41-s − 344·43-s − 238·45-s − 84·47-s − 438·53-s + 448·55-s + 56·59-s − 98·61-s + 392·65-s + 336·67-s + 896·71-s − 966·73-s − 52·79-s − 440·81-s − 392·83-s + 2.15e3·85-s + ⋯ |
| L(s) = 1 | + 1.25·5-s − 0.629·9-s + 0.877·11-s + 0.597·13-s + 2.19·17-s + 2.70·19-s + 0.616·23-s + 9/25·25-s − 1.51·29-s + 1.13·31-s + 1.53·37-s + 1.59·41-s − 1.21·43-s − 0.788·45-s − 0.260·47-s − 1.13·53-s + 1.09·55-s + 0.123·59-s − 0.205·61-s + 0.748·65-s + 0.612·67-s + 1.49·71-s − 1.54·73-s − 0.0740·79-s − 0.603·81-s − 0.518·83-s + 2.75·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38416 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.031318495\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.031318495\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 17 T^{2} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 14 T + 151 T^{2} - 14 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 32 T + 1105 T^{2} - 32 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 28 T + 3998 T^{2} - 28 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 154 T + 15163 T^{2} - 154 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 224 T + 25929 T^{2} - 224 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 68 T + 23677 T^{2} - 68 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 236 T + 33694 T^{2} + 236 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 196 T + 67373 T^{2} - 196 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 346 T + 65967 T^{2} - 346 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 420 T + 176614 T^{2} - 420 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 p T + p^{3} T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 84 T + 201085 T^{2} + 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 438 T + 280447 T^{2} + 438 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 56 T + 360889 T^{2} - 56 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 98 T + 253751 T^{2} + 98 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 336 T + 627937 T^{2} - 336 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 896 T + 800494 T^{2} - 896 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 966 T + 187 p^{2} T^{2} + 966 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 52 T + 332261 T^{2} + 52 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 392 T + 895462 T^{2} + 392 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 294 T + 1298347 T^{2} + 294 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 420 T + 1655734 T^{2} - 420 p^{3} T^{3} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.05613945590060643502268671308, −11.90777149684239752256777139971, −11.29532297672240342727652536051, −10.97841572020270565337340240016, −10.06429052803319887777105325395, −9.672585717999712280132310616285, −9.504590003257297971959538773991, −9.078576199380446086013606393141, −8.066054946025792455243967397498, −7.86907994128073518564882708202, −7.17278286886916339240778986758, −6.46456560401594350202960942345, −5.75103747986384651107101865134, −5.63838515262862702990974097871, −5.05684794307818955601177512364, −3.95556910916878174396278543679, −3.22458729042098121325360950286, −2.76078108111994733835087058823, −1.39120033214515886397948951267, −1.07761830971109886104970621694,
1.07761830971109886104970621694, 1.39120033214515886397948951267, 2.76078108111994733835087058823, 3.22458729042098121325360950286, 3.95556910916878174396278543679, 5.05684794307818955601177512364, 5.63838515262862702990974097871, 5.75103747986384651107101865134, 6.46456560401594350202960942345, 7.17278286886916339240778986758, 7.86907994128073518564882708202, 8.066054946025792455243967397498, 9.078576199380446086013606393141, 9.504590003257297971959538773991, 9.672585717999712280132310616285, 10.06429052803319887777105325395, 10.97841572020270565337340240016, 11.29532297672240342727652536051, 11.90777149684239752256777139971, 12.05613945590060643502268671308
