| L(s) = 1 | − 4·2-s + 14·3-s + 16·4-s − 45·5-s − 56·6-s − 121·7-s − 64·8-s − 47·9-s + 180·10-s − 381·11-s + 224·12-s + 100·13-s + 484·14-s − 630·15-s + 256·16-s + 933·17-s + 188·18-s − 720·20-s − 1.69e3·21-s + 1.52e3·22-s − 552·23-s − 896·24-s − 1.10e3·25-s − 400·26-s − 4.06e3·27-s − 1.93e3·28-s − 2.39e3·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.898·3-s + 1/2·4-s − 0.804·5-s − 0.635·6-s − 0.933·7-s − 0.353·8-s − 0.193·9-s + 0.569·10-s − 0.949·11-s + 0.449·12-s + 0.164·13-s + 0.659·14-s − 0.722·15-s + 1/4·16-s + 0.782·17-s + 0.136·18-s − 0.402·20-s − 0.838·21-s + 0.671·22-s − 0.217·23-s − 0.317·24-s − 0.351·25-s − 0.116·26-s − 1.07·27-s − 0.466·28-s − 0.528·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.7091059937\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7091059937\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{2} T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 14 T + p^{5} T^{2} \) |
| 5 | \( 1 + 9 p T + p^{5} T^{2} \) |
| 7 | \( 1 + 121 T + p^{5} T^{2} \) |
| 11 | \( 1 + 381 T + p^{5} T^{2} \) |
| 13 | \( 1 - 100 T + p^{5} T^{2} \) |
| 17 | \( 1 - 933 T + p^{5} T^{2} \) |
| 23 | \( 1 + 24 p T + p^{5} T^{2} \) |
| 29 | \( 1 + 2394 T + p^{5} T^{2} \) |
| 31 | \( 1 - 4024 T + p^{5} T^{2} \) |
| 37 | \( 1 + 9182 T + p^{5} T^{2} \) |
| 41 | \( 1 - 2250 T + p^{5} T^{2} \) |
| 43 | \( 1 + 23377 T + p^{5} T^{2} \) |
| 47 | \( 1 + 26595 T + p^{5} T^{2} \) |
| 53 | \( 1 - 16008 T + p^{5} T^{2} \) |
| 59 | \( 1 - 126 T + p^{5} T^{2} \) |
| 61 | \( 1 - 21335 T + p^{5} T^{2} \) |
| 67 | \( 1 - 51760 T + p^{5} T^{2} \) |
| 71 | \( 1 + 8574 T + p^{5} T^{2} \) |
| 73 | \( 1 - 11153 T + p^{5} T^{2} \) |
| 79 | \( 1 - 1660 T + p^{5} T^{2} \) |
| 83 | \( 1 - 95964 T + p^{5} T^{2} \) |
| 89 | \( 1 + 118848 T + p^{5} T^{2} \) |
| 97 | \( 1 - 153760 T + p^{5} T^{2} \) |
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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
−9.695407131269886716994809848164, −8.572106560408839510955122933511, −8.116396542456960320270530881079, −7.37414211517938140137337941977, −6.35465660342733497296581277465, −5.21350568950759901352590520898, −3.60755679523668900965069214290, −3.13958851128319633850481555784, −2.00495178789845608762045353915, −0.39335303336151407056024564617,
0.39335303336151407056024564617, 2.00495178789845608762045353915, 3.13958851128319633850481555784, 3.60755679523668900965069214290, 5.21350568950759901352590520898, 6.35465660342733497296581277465, 7.37414211517938140137337941977, 8.116396542456960320270530881079, 8.572106560408839510955122933511, 9.695407131269886716994809848164
