L(s) = 1 | − 3-s − 4·5-s + 2·7-s − 2·9-s − 4·11-s − 5·13-s + 4·15-s − 2·17-s + 6·19-s − 2·21-s + 23-s + 11·25-s + 5·27-s + 29-s − 9·31-s + 4·33-s − 8·35-s − 4·37-s + 5·39-s + 3·41-s + 8·43-s + 8·45-s − 5·47-s − 3·49-s + 2·51-s + 6·53-s + 16·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.78·5-s + 0.755·7-s − 2/3·9-s − 1.20·11-s − 1.38·13-s + 1.03·15-s − 0.485·17-s + 1.37·19-s − 0.436·21-s + 0.208·23-s + 11/5·25-s + 0.962·27-s + 0.185·29-s − 1.61·31-s + 0.696·33-s − 1.35·35-s − 0.657·37-s + 0.800·39-s + 0.468·41-s + 1.21·43-s + 1.19·45-s − 0.729·47-s − 3/7·49-s + 0.280·51-s + 0.824·53-s + 2.15·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 5 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 + 15 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−11.91079007041406289824271502817, −11.32872114110074272432097103072, −10.53801091148756483915176434884, −8.910132789738256271513641474783, −7.74681990799276192851943075763, −7.33683533600380364349770730667, −5.38734001699377202464167768623, −4.60774849630753565059580112097, −2.99122119577828822953767314363, 0,
2.99122119577828822953767314363, 4.60774849630753565059580112097, 5.38734001699377202464167768623, 7.33683533600380364349770730667, 7.74681990799276192851943075763, 8.910132789738256271513641474783, 10.53801091148756483915176434884, 11.32872114110074272432097103072, 11.91079007041406289824271502817