| L(s) = 1 | − 2·2-s + 8·3-s + 4·4-s − 16·6-s + 4·7-s − 8·8-s + 37·9-s + 12·11-s + 32·12-s + 58·13-s − 8·14-s + 16·16-s − 66·17-s − 74·18-s − 100·19-s + 32·21-s − 24·22-s − 132·23-s − 64·24-s − 116·26-s + 80·27-s + 16·28-s − 90·29-s + 152·31-s − 32·32-s + 96·33-s + 132·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.53·3-s + 1/2·4-s − 1.08·6-s + 0.215·7-s − 0.353·8-s + 1.37·9-s + 0.328·11-s + 0.769·12-s + 1.23·13-s − 0.152·14-s + 1/4·16-s − 0.941·17-s − 0.968·18-s − 1.20·19-s + 0.332·21-s − 0.232·22-s − 1.19·23-s − 0.544·24-s − 0.874·26-s + 0.570·27-s + 0.107·28-s − 0.576·29-s + 0.880·31-s − 0.176·32-s + 0.506·33-s + 0.665·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.578565699\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.578565699\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 8 T + p^{3} T^{2} \) |
| 7 | \( 1 - 4 T + p^{3} T^{2} \) |
| 11 | \( 1 - 12 T + p^{3} T^{2} \) |
| 13 | \( 1 - 58 T + p^{3} T^{2} \) |
| 17 | \( 1 + 66 T + p^{3} T^{2} \) |
| 19 | \( 1 + 100 T + p^{3} T^{2} \) |
| 23 | \( 1 + 132 T + p^{3} T^{2} \) |
| 29 | \( 1 + 90 T + p^{3} T^{2} \) |
| 31 | \( 1 - 152 T + p^{3} T^{2} \) |
| 37 | \( 1 - 34 T + p^{3} T^{2} \) |
| 41 | \( 1 + 438 T + p^{3} T^{2} \) |
| 43 | \( 1 + 32 T + p^{3} T^{2} \) |
| 47 | \( 1 - 204 T + p^{3} T^{2} \) |
| 53 | \( 1 + 222 T + p^{3} T^{2} \) |
| 59 | \( 1 - 420 T + p^{3} T^{2} \) |
| 61 | \( 1 - 902 T + p^{3} T^{2} \) |
| 67 | \( 1 - 1024 T + p^{3} T^{2} \) |
| 71 | \( 1 - 432 T + p^{3} T^{2} \) |
| 73 | \( 1 + 362 T + p^{3} T^{2} \) |
| 79 | \( 1 + 160 T + p^{3} T^{2} \) |
| 83 | \( 1 + 72 T + p^{3} T^{2} \) |
| 89 | \( 1 - 810 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1106 T + p^{3} T^{2} \) |
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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
−15.10527647695033303834171950661, −14.03666354063861339240147928179, −13.03471815163085802663623650894, −11.33448589653740591437742921768, −9.975023505642710053505058933239, −8.716638835363535290756995180948, −8.192325937919748535755327852198, −6.57235269598188037993838794931, −3.84058546664873519083326793824, −2.04250149809618508378100381796,
2.04250149809618508378100381796, 3.84058546664873519083326793824, 6.57235269598188037993838794931, 8.192325937919748535755327852198, 8.716638835363535290756995180948, 9.975023505642710053505058933239, 11.33448589653740591437742921768, 13.03471815163085802663623650894, 14.03666354063861339240147928179, 15.10527647695033303834171950661
