| L(s) = 1 | − 3-s − 2·9-s − 2·11-s + 13-s − 2·17-s + 8·19-s + 23-s + 5·27-s + 9·29-s − 4·31-s + 2·33-s − 6·37-s − 39-s − 2·41-s − 9·43-s − 4·47-s − 7·49-s + 2·51-s − 13·53-s − 8·57-s + 6·59-s − 5·61-s − 6·67-s − 69-s + 2·71-s − 8·73-s + 17·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 2/3·9-s − 0.603·11-s + 0.277·13-s − 0.485·17-s + 1.83·19-s + 0.208·23-s + 0.962·27-s + 1.67·29-s − 0.718·31-s + 0.348·33-s − 0.986·37-s − 0.160·39-s − 0.312·41-s − 1.37·43-s − 0.583·47-s − 49-s + 0.280·51-s − 1.78·53-s − 1.05·57-s + 0.781·59-s − 0.640·61-s − 0.733·67-s − 0.120·69-s + 0.237·71-s − 0.936·73-s + 1.91·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 13 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 17 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−8.437171076984545082345925716006, −7.80736393935212151429199153301, −6.82283932019898209582572698469, −6.21379395933571995843771342229, −5.17243840517550458936581582840, −4.94070877587431892381858861851, −3.46823434384602215691515354309, −2.80420907264271587334604654426, −1.38884893432392505102265384371, 0,
1.38884893432392505102265384371, 2.80420907264271587334604654426, 3.46823434384602215691515354309, 4.94070877587431892381858861851, 5.17243840517550458936581582840, 6.21379395933571995843771342229, 6.82283932019898209582572698469, 7.80736393935212151429199153301, 8.437171076984545082345925716006
