L(s) = 1 | − 2-s + 4-s − 4·5-s + 7-s − 8-s + 4·10-s + 2·11-s − 2·13-s − 14-s + 16-s + 19-s − 4·20-s − 2·22-s + 2·23-s + 11·25-s + 2·26-s + 28-s − 6·29-s + 4·31-s − 32-s − 4·35-s + 2·37-s − 38-s + 4·40-s − 10·41-s + 4·43-s + 2·44-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.78·5-s + 0.377·7-s − 0.353·8-s + 1.26·10-s + 0.603·11-s − 0.554·13-s − 0.267·14-s + 1/4·16-s + 0.229·19-s − 0.894·20-s − 0.426·22-s + 0.417·23-s + 11/5·25-s + 0.392·26-s + 0.188·28-s − 1.11·29-s + 0.718·31-s − 0.176·32-s − 0.676·35-s + 0.328·37-s − 0.162·38-s + 0.632·40-s − 1.56·41-s + 0.609·43-s + 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 - T \) |
good | 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−8.556090712772739088663238459301, −7.77828121628680667033885713608, −7.32424551774384528725879670738, −6.62104491989591150138784116150, −5.37541185469873861025392802404, −4.40472302626164590382346795724, −3.69920031178742845346911959014, −2.70799792335600379572107311013, −1.24539258123918074733585298308, 0,
1.24539258123918074733585298308, 2.70799792335600379572107311013, 3.69920031178742845346911959014, 4.40472302626164590382346795724, 5.37541185469873861025392802404, 6.62104491989591150138784116150, 7.32424551774384528725879670738, 7.77828121628680667033885713608, 8.556090712772739088663238459301