| L(s) = 1 | − 8·4-s + 12·5-s + 2·7-s + 36·11-s + 13·13-s + 64·16-s + 78·17-s + 74·19-s − 96·20-s + 96·23-s + 19·25-s − 16·28-s − 18·29-s − 214·31-s + 24·35-s − 286·37-s + 384·41-s + 524·43-s − 288·44-s − 300·47-s − 339·49-s − 104·52-s − 558·53-s + 432·55-s − 576·59-s + 74·61-s − 512·64-s + ⋯ |
| L(s) = 1 | − 4-s + 1.07·5-s + 0.107·7-s + 0.986·11-s + 0.277·13-s + 16-s + 1.11·17-s + 0.893·19-s − 1.07·20-s + 0.870·23-s + 0.151·25-s − 0.107·28-s − 0.115·29-s − 1.23·31-s + 0.115·35-s − 1.27·37-s + 1.46·41-s + 1.85·43-s − 0.986·44-s − 0.931·47-s − 0.988·49-s − 0.277·52-s − 1.44·53-s + 1.05·55-s − 1.27·59-s + 0.155·61-s − 64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.664545579\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.664545579\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 - p T \) |
| good | 2 | \( 1 + p^{3} T^{2} \) |
| 5 | \( 1 - 12 T + p^{3} T^{2} \) |
| 7 | \( 1 - 2 T + p^{3} T^{2} \) |
| 11 | \( 1 - 36 T + p^{3} T^{2} \) |
| 17 | \( 1 - 78 T + p^{3} T^{2} \) |
| 19 | \( 1 - 74 T + p^{3} T^{2} \) |
| 23 | \( 1 - 96 T + p^{3} T^{2} \) |
| 29 | \( 1 + 18 T + p^{3} T^{2} \) |
| 31 | \( 1 + 214 T + p^{3} T^{2} \) |
| 37 | \( 1 + 286 T + p^{3} T^{2} \) |
| 41 | \( 1 - 384 T + p^{3} T^{2} \) |
| 43 | \( 1 - 524 T + p^{3} T^{2} \) |
| 47 | \( 1 + 300 T + p^{3} T^{2} \) |
| 53 | \( 1 + 558 T + p^{3} T^{2} \) |
| 59 | \( 1 + 576 T + p^{3} T^{2} \) |
| 61 | \( 1 - 74 T + p^{3} T^{2} \) |
| 67 | \( 1 - 38 T + p^{3} T^{2} \) |
| 71 | \( 1 - 456 T + p^{3} T^{2} \) |
| 73 | \( 1 + 682 T + p^{3} T^{2} \) |
| 79 | \( 1 - 704 T + p^{3} T^{2} \) |
| 83 | \( 1 - 888 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1020 T + p^{3} T^{2} \) |
| 97 | \( 1 - 110 T + p^{3} T^{2} \) |
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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
−13.16764050290785923290959247631, −12.24393836531819647663790523192, −10.81986235342614299045512697854, −9.533879106083709872149126257382, −9.171890874523769751163248199965, −7.67355073251042162816905051866, −6.07544632827492978542972904067, −5.08538833271391436376712473514, −3.51780047655492462589364685314, −1.30527217470593149414341073126,
1.30527217470593149414341073126, 3.51780047655492462589364685314, 5.08538833271391436376712473514, 6.07544632827492978542972904067, 7.67355073251042162816905051866, 9.171890874523769751163248199965, 9.533879106083709872149126257382, 10.81986235342614299045512697854, 12.24393836531819647663790523192, 13.16764050290785923290959247631
