| L(s) = 1 | − 3·3-s − 5·5-s − 2·7-s + 9·9-s − 30·11-s − 2·13-s + 15·15-s − 54·17-s + 106·19-s + 6·21-s + 18·23-s + 25·25-s − 27·27-s + 138·29-s − 292·31-s + 90·33-s + 10·35-s − 270·37-s + 6·39-s − 466·41-s − 32·43-s − 45·45-s + 74·47-s − 339·49-s + 162·51-s + 302·53-s + 150·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.107·7-s + 1/3·9-s − 0.822·11-s − 0.0426·13-s + 0.258·15-s − 0.770·17-s + 1.27·19-s + 0.0623·21-s + 0.163·23-s + 1/5·25-s − 0.192·27-s + 0.883·29-s − 1.69·31-s + 0.474·33-s + 0.0482·35-s − 1.19·37-s + 0.0246·39-s − 1.77·41-s − 0.113·43-s − 0.149·45-s + 0.229·47-s − 0.988·49-s + 0.444·51-s + 0.782·53-s + 0.367·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9037386784\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9037386784\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 + p T \) |
| good | 7 | \( 1 + 2 T + p^{3} T^{2} \) |
| 11 | \( 1 + 30 T + p^{3} T^{2} \) |
| 13 | \( 1 + 2 T + p^{3} T^{2} \) |
| 17 | \( 1 + 54 T + p^{3} T^{2} \) |
| 19 | \( 1 - 106 T + p^{3} T^{2} \) |
| 23 | \( 1 - 18 T + p^{3} T^{2} \) |
| 29 | \( 1 - 138 T + p^{3} T^{2} \) |
| 31 | \( 1 + 292 T + p^{3} T^{2} \) |
| 37 | \( 1 + 270 T + p^{3} T^{2} \) |
| 41 | \( 1 + 466 T + p^{3} T^{2} \) |
| 43 | \( 1 + 32 T + p^{3} T^{2} \) |
| 47 | \( 1 - 74 T + p^{3} T^{2} \) |
| 53 | \( 1 - 302 T + p^{3} T^{2} \) |
| 59 | \( 1 - 518 T + p^{3} T^{2} \) |
| 61 | \( 1 + 86 T + p^{3} T^{2} \) |
| 67 | \( 1 + 448 T + p^{3} T^{2} \) |
| 71 | \( 1 - 328 T + p^{3} T^{2} \) |
| 73 | \( 1 - 258 T + p^{3} T^{2} \) |
| 79 | \( 1 - 288 T + p^{3} T^{2} \) |
| 83 | \( 1 - 236 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1254 T + p^{3} T^{2} \) |
| 97 | \( 1 + 790 T + p^{3} T^{2} \) |
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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
−8.813483063614097419652865338111, −8.009011128935705626579160788328, −7.17434496401149581644720713841, −6.61669316684672375972116723161, −5.39411828641656510260136310502, −5.03660618731131770261732604316, −3.87856512236311230754446885343, −3.00250349855259251295587129156, −1.75524469938146499941777608180, −0.44846442092585047321767292918,
0.44846442092585047321767292918, 1.75524469938146499941777608180, 3.00250349855259251295587129156, 3.87856512236311230754446885343, 5.03660618731131770261732604316, 5.39411828641656510260136310502, 6.61669316684672375972116723161, 7.17434496401149581644720713841, 8.009011128935705626579160788328, 8.813483063614097419652865338111
