| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s − 2·11-s − 12-s + 13-s + 16-s + 5·17-s − 18-s − 2·19-s + 2·22-s + 4·23-s + 24-s − 26-s − 27-s − 8·29-s − 7·31-s − 32-s + 2·33-s − 5·34-s + 36-s + 37-s + 2·38-s − 39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.603·11-s − 0.288·12-s + 0.277·13-s + 1/4·16-s + 1.21·17-s − 0.235·18-s − 0.458·19-s + 0.426·22-s + 0.834·23-s + 0.204·24-s − 0.196·26-s − 0.192·27-s − 1.48·29-s − 1.25·31-s − 0.176·32-s + 0.348·33-s − 0.857·34-s + 1/6·36-s + 0.164·37-s + 0.324·38-s − 0.160·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 + 15 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−7.75081125011270975893010095533, −7.20953663653728906505807418705, −6.45964267034518695697473416742, −5.55715225970058464984097030591, −5.23052982012440010740442139080, −3.99882993402388527352693721413, −3.21755261030788370517639153059, −2.12910702153576203643555568356, −1.16758049153403985778523550895, 0,
1.16758049153403985778523550895, 2.12910702153576203643555568356, 3.21755261030788370517639153059, 3.99882993402388527352693721413, 5.23052982012440010740442139080, 5.55715225970058464984097030591, 6.45964267034518695697473416742, 7.20953663653728906505807418705, 7.75081125011270975893010095533
