| L(s) = 1 | + 2-s − 4-s − 4·5-s − 2·7-s − 3·8-s − 4·10-s − 11-s + 2·13-s − 2·14-s − 16-s + 2·17-s + 4·20-s − 22-s + 4·23-s + 11·25-s + 2·26-s + 2·28-s + 6·29-s − 4·31-s + 5·32-s + 2·34-s + 8·35-s + 6·37-s + 12·40-s + 10·41-s + 6·43-s + 44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.78·5-s − 0.755·7-s − 1.06·8-s − 1.26·10-s − 0.301·11-s + 0.554·13-s − 0.534·14-s − 1/4·16-s + 0.485·17-s + 0.894·20-s − 0.213·22-s + 0.834·23-s + 11/5·25-s + 0.392·26-s + 0.377·28-s + 1.11·29-s − 0.718·31-s + 0.883·32-s + 0.342·34-s + 1.35·35-s + 0.986·37-s + 1.89·40-s + 1.56·41-s + 0.914·43-s + 0.150·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35739 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35739 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.084229574\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.084229574\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.72393643099934, −14.70687924372856, −13.76909702692379, −13.30883891383936, −12.76431322104590, −12.29638852316109, −12.09675164382289, −11.10572124277904, −11.06769748675466, −10.21515768683207, −9.377926069982859, −9.102612808731882, −8.359710444987837, −7.874429541842715, −7.440545767660799, −6.580401044067281, −6.185466129732394, −5.353600123068065, −4.760742005533437, −4.177011132503010, −3.763756264260263, −3.040865314691575, −2.805939622714208, −1.128517112650661, −0.4079444885027822,
0.4079444885027822, 1.128517112650661, 2.805939622714208, 3.040865314691575, 3.763756264260263, 4.177011132503010, 4.760742005533437, 5.353600123068065, 6.185466129732394, 6.580401044067281, 7.440545767660799, 7.874429541842715, 8.359710444987837, 9.102612808731882, 9.377926069982859, 10.21515768683207, 11.06769748675466, 11.10572124277904, 12.09675164382289, 12.29638852316109, 12.76431322104590, 13.30883891383936, 13.76909702692379, 14.70687924372856, 14.72393643099934
