| L(s) = 1 | + 3-s − 2·7-s − 2·9-s − 13-s + 6·17-s − 2·19-s − 2·21-s − 5·25-s − 5·27-s − 3·29-s + 5·31-s − 8·37-s − 39-s + 3·41-s − 8·43-s + 9·47-s − 3·49-s + 6·51-s − 6·53-s − 2·57-s − 12·59-s − 14·61-s + 4·63-s − 8·67-s − 15·71-s − 7·73-s − 5·75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.755·7-s − 2/3·9-s − 0.277·13-s + 1.45·17-s − 0.458·19-s − 0.436·21-s − 25-s − 0.962·27-s − 0.557·29-s + 0.898·31-s − 1.31·37-s − 0.160·39-s + 0.468·41-s − 1.21·43-s + 1.31·47-s − 3/7·49-s + 0.840·51-s − 0.824·53-s − 0.264·57-s − 1.56·59-s − 1.79·61-s + 0.503·63-s − 0.977·67-s − 1.78·71-s − 0.819·73-s − 0.577·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2116 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 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
−8.764372579299956656234992702503, −7.910283532428831686026568439904, −7.36215540469752961762928495977, −6.20382398868342034524355070702, −5.71274111334034856450397667213, −4.58604999215258623252929989753, −3.43650325767803320317918790111, −2.97513946462044282901176605228, −1.74242190941620479556994328523, 0,
1.74242190941620479556994328523, 2.97513946462044282901176605228, 3.43650325767803320317918790111, 4.58604999215258623252929989753, 5.71274111334034856450397667213, 6.20382398868342034524355070702, 7.36215540469752961762928495977, 7.910283532428831686026568439904, 8.764372579299956656234992702503
