| L(s) = 1 | − 6·2-s − 3-s + 24·4-s + 15·5-s + 6·6-s + 7·7-s − 80·8-s − 35·9-s − 90·10-s + 27·11-s − 24·12-s + 75·13-s − 42·14-s − 15·15-s + 240·16-s + 127·17-s + 210·18-s − 185·19-s + 360·20-s − 7·21-s − 162·22-s + 69·23-s + 80·24-s + 150·25-s − 450·26-s + 77·27-s + 168·28-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 0.192·3-s + 3·4-s + 1.34·5-s + 0.408·6-s + 0.377·7-s − 3.53·8-s − 1.29·9-s − 2.84·10-s + 0.740·11-s − 0.577·12-s + 1.60·13-s − 0.801·14-s − 0.258·15-s + 15/4·16-s + 1.81·17-s + 2.74·18-s − 2.23·19-s + 4.02·20-s − 0.0727·21-s − 1.56·22-s + 0.625·23-s + 0.680·24-s + 6/5·25-s − 3.39·26-s + 0.548·27-s + 1.13·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12167000 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12167000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.222299582\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.222299582\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{3} \) |
| 5 | $C_1$ | \( ( 1 - p T )^{3} \) |
| 23 | $C_1$ | \( ( 1 - p T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 + T + 4 p^{2} T^{2} - 2 p T^{3} + 4 p^{5} T^{4} + p^{6} T^{5} + p^{9} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - p T + 100 T^{2} - 2234 T^{3} + 100 p^{3} T^{4} - p^{7} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 27 T + 54 p T^{2} + 17514 T^{3} + 54 p^{4} T^{4} - 27 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 75 T + 2928 T^{2} - 54312 T^{3} + 2928 p^{3} T^{4} - 75 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 127 T + 14630 T^{2} - 1151352 T^{3} + 14630 p^{3} T^{4} - 127 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 185 T + 27580 T^{2} + 2575426 T^{3} + 27580 p^{3} T^{4} + 185 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 344 T + 89719 T^{2} - 16488344 T^{3} + 89719 p^{3} T^{4} - 344 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 397 T + 115538 T^{2} - 22106974 T^{3} + 115538 p^{3} T^{4} - 397 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 978 T + 465279 T^{2} - 132082804 T^{3} + 465279 p^{3} T^{4} - 978 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 575 T + 150520 T^{2} + 28305272 T^{3} + 150520 p^{3} T^{4} + 575 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 812 T + 379001 T^{2} - 118957448 T^{3} + 379001 p^{3} T^{4} - 812 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 270 T + 238809 T^{2} + 59248420 T^{3} + 238809 p^{3} T^{4} + 270 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 510 T + 273939 T^{2} - 62350836 T^{3} + 273939 p^{3} T^{4} - 510 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 142 T + 479357 T^{2} - 68234340 T^{3} + 479357 p^{3} T^{4} - 142 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 49 T + 528970 T^{2} - 1328020 T^{3} + 528970 p^{3} T^{4} + 49 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 1616 T + 24987 p T^{2} - 1083666976 T^{3} + 24987 p^{4} T^{4} - 1616 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 471 T + 915408 T^{2} + 261548402 T^{3} + 915408 p^{3} T^{4} + 471 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 780 T + 315339 T^{2} - 65997376 T^{3} + 315339 p^{3} T^{4} + 780 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 860 T + 1547325 T^{2} + 839730376 T^{3} + 1547325 p^{3} T^{4} + 860 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 288 T + 15879 p T^{2} + 223172720 T^{3} + 15879 p^{4} T^{4} + 288 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 90 T - 1989 p T^{2} - 983003148 T^{3} - 1989 p^{4} T^{4} + 90 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 1321 T + 2369566 T^{2} - 1721904884 T^{3} + 2369566 p^{3} T^{4} - 1321 p^{6} T^{5} + p^{9} T^{6} \) |
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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
−10.26264198333893455563637506200, −10.14371333386047858143649086692, −9.754437602719335880740009251598, −9.691820733293265087394706184069, −8.856183241906944430418531394633, −8.781708415186736967016304768745, −8.768205236052827149289716323440, −8.141848119525796217840374533212, −7.984413571632123416122587518040, −7.87477636028186694918466906435, −6.79333650796890280496185443296, −6.78879163010938414608756163439, −6.41428386120771470466303122723, −5.97865185691510911930799936709, −5.86931720952461481216839434665, −5.61057214595317156839837665673, −4.60101127242223195463934339678, −4.47427118315126127356962711490, −3.60807349797765988781896815630, −2.94798691933596377241175459452, −2.48331950479038644699031082063, −2.41525348158693546969786872192, −1.23754612585730579670345808622, −1.10780033761726753687201787193, −0.72091258386962182676758978260,
0.72091258386962182676758978260, 1.10780033761726753687201787193, 1.23754612585730579670345808622, 2.41525348158693546969786872192, 2.48331950479038644699031082063, 2.94798691933596377241175459452, 3.60807349797765988781896815630, 4.47427118315126127356962711490, 4.60101127242223195463934339678, 5.61057214595317156839837665673, 5.86931720952461481216839434665, 5.97865185691510911930799936709, 6.41428386120771470466303122723, 6.78879163010938414608756163439, 6.79333650796890280496185443296, 7.87477636028186694918466906435, 7.984413571632123416122587518040, 8.141848119525796217840374533212, 8.768205236052827149289716323440, 8.781708415186736967016304768745, 8.856183241906944430418531394633, 9.691820733293265087394706184069, 9.754437602719335880740009251598, 10.14371333386047858143649086692, 10.26264198333893455563637506200
