| L(s) = 1 | − 2-s + 4-s − 5-s + 7-s − 8-s + 10-s − 6·13-s − 14-s + 16-s + 6·17-s + 4·19-s − 20-s − 23-s + 25-s + 6·26-s + 28-s + 6·29-s − 32-s − 6·34-s − 35-s − 10·37-s − 4·38-s + 40-s − 2·41-s − 8·43-s + 46-s − 8·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s − 0.353·8-s + 0.316·10-s − 1.66·13-s − 0.267·14-s + 1/4·16-s + 1.45·17-s + 0.917·19-s − 0.223·20-s − 0.208·23-s + 1/5·25-s + 1.17·26-s + 0.188·28-s + 1.11·29-s − 0.176·32-s − 1.02·34-s − 0.169·35-s − 1.64·37-s − 0.648·38-s + 0.158·40-s − 0.312·41-s − 1.21·43-s + 0.147·46-s − 1.16·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−16.29795470682488, −16.09523816091654, −15.21681436291009, −14.80031234029041, −14.29932100230773, −13.75273508731428, −12.82965745213954, −12.20356035308485, −11.77690140244267, −11.54324844825783, −10.39598219942909, −10.09279290375249, −9.725260543958465, −8.781813214353518, −8.296755880301810, −7.694925562615755, −7.160297300774111, −6.736191904684373, −5.544826780484512, −5.212512022177679, −4.424929880246081, −3.400844924868045, −2.882885896847072, −1.916072952045330, −1.050794726231428, 0,
1.050794726231428, 1.916072952045330, 2.882885896847072, 3.400844924868045, 4.424929880246081, 5.212512022177679, 5.544826780484512, 6.736191904684373, 7.160297300774111, 7.694925562615755, 8.296755880301810, 8.781813214353518, 9.725260543958465, 10.09279290375249, 10.39598219942909, 11.54324844825783, 11.77690140244267, 12.20356035308485, 12.82965745213954, 13.75273508731428, 14.29932100230773, 14.80031234029041, 15.21681436291009, 16.09523816091654, 16.29795470682488
