| L(s) = 1 | + 3·3-s + 3.18·5-s + 23.9·7-s + 9·9-s − 6.74·11-s + 13·13-s + 9.55·15-s + 104.·17-s − 137.·19-s + 71.8·21-s + 110.·23-s − 114.·25-s + 27·27-s − 57.6·29-s + 319.·31-s − 20.2·33-s + 76.2·35-s − 2.88·37-s + 39·39-s + 319.·41-s + 344.·43-s + 28.6·45-s − 439.·47-s + 229.·49-s + 312.·51-s − 97.2·53-s − 21.5·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.285·5-s + 1.29·7-s + 0.333·9-s − 0.184·11-s + 0.277·13-s + 0.164·15-s + 1.48·17-s − 1.66·19-s + 0.746·21-s + 0.999·23-s − 0.918·25-s + 0.192·27-s − 0.368·29-s + 1.85·31-s − 0.106·33-s + 0.368·35-s − 0.0128·37-s + 0.160·39-s + 1.21·41-s + 1.22·43-s + 0.0950·45-s − 1.36·47-s + 0.670·49-s + 0.858·51-s − 0.252·53-s − 0.0527·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.821722039\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.821722039\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 13 | \( 1 - 13T \) |
| good | 5 | \( 1 - 3.18T + 125T^{2} \) |
| 7 | \( 1 - 23.9T + 343T^{2} \) |
| 11 | \( 1 + 6.74T + 1.33e3T^{2} \) |
| 17 | \( 1 - 104.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 137.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 110.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 57.6T + 2.43e4T^{2} \) |
| 31 | \( 1 - 319.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 2.88T + 5.06e4T^{2} \) |
| 41 | \( 1 - 319.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 344.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 439.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 97.2T + 1.48e5T^{2} \) |
| 59 | \( 1 - 448.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 264.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 712.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.13e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 666.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 828.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 734.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 153.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 569.T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.15003699293305770209381085460, −10.34327466307908007684873644831, −9.298997621017830567667931692679, −8.231348745215258846556215206362, −7.74625267847375434307166331133, −6.32463786780079316901959010933, −5.13008107496469887309850665877, −4.05860914379598049431676422649, −2.55081009997226885584129771962, −1.29496283448469993975045199689,
1.29496283448469993975045199689, 2.55081009997226885584129771962, 4.05860914379598049431676422649, 5.13008107496469887309850665877, 6.32463786780079316901959010933, 7.74625267847375434307166331133, 8.231348745215258846556215206362, 9.298997621017830567667931692679, 10.34327466307908007684873644831, 11.15003699293305770209381085460
