| L(s) = 1 | − 2-s + 0.479·3-s + 4-s − 2.85·5-s − 0.479·6-s + 7-s − 8-s − 2.76·9-s + 2.85·10-s + 5.13·11-s + 0.479·12-s − 0.235·13-s − 14-s − 1.37·15-s + 16-s − 4.72·17-s + 2.76·18-s − 2.85·20-s + 0.479·21-s − 5.13·22-s + 6.09·23-s − 0.479·24-s + 3.15·25-s + 0.235·26-s − 2.76·27-s + 28-s − 5.85·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.277·3-s + 0.5·4-s − 1.27·5-s − 0.195·6-s + 0.377·7-s − 0.353·8-s − 0.923·9-s + 0.902·10-s + 1.54·11-s + 0.138·12-s − 0.0652·13-s − 0.267·14-s − 0.353·15-s + 0.250·16-s − 1.14·17-s + 0.652·18-s − 0.638·20-s + 0.104·21-s − 1.09·22-s + 1.27·23-s − 0.0979·24-s + 0.630·25-s + 0.0461·26-s − 0.532·27-s + 0.188·28-s − 1.08·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 0.479T + 3T^{2} \) |
| 5 | \( 1 + 2.85T + 5T^{2} \) |
| 11 | \( 1 - 5.13T + 11T^{2} \) |
| 13 | \( 1 + 0.235T + 13T^{2} \) |
| 17 | \( 1 + 4.72T + 17T^{2} \) |
| 23 | \( 1 - 6.09T + 23T^{2} \) |
| 29 | \( 1 + 5.85T + 29T^{2} \) |
| 31 | \( 1 - 5.68T + 31T^{2} \) |
| 37 | \( 1 + 2.94T + 37T^{2} \) |
| 41 | \( 1 - 3.33T + 41T^{2} \) |
| 43 | \( 1 + 6.50T + 43T^{2} \) |
| 47 | \( 1 + 0.589T + 47T^{2} \) |
| 53 | \( 1 - 8.65T + 53T^{2} \) |
| 59 | \( 1 - 8.79T + 59T^{2} \) |
| 61 | \( 1 + 5.43T + 61T^{2} \) |
| 67 | \( 1 + 11.1T + 67T^{2} \) |
| 71 | \( 1 + 3.04T + 71T^{2} \) |
| 73 | \( 1 - 10.8T + 73T^{2} \) |
| 79 | \( 1 + 10.2T + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 + 15.2T + 89T^{2} \) |
| 97 | \( 1 + 10.2T + 97T^{2} \) |
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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
−8.039235058620029408235004595080, −7.20928811675510186860511007909, −6.75883960639560868450780085036, −5.86236360303133458375920097279, −4.78790728346015599888920755649, −3.99065861828001880082134384176, −3.32285321306006299571845873218, −2.33613291328013782488075808886, −1.17965341171357015550276668293, 0,
1.17965341171357015550276668293, 2.33613291328013782488075808886, 3.32285321306006299571845873218, 3.99065861828001880082134384176, 4.78790728346015599888920755649, 5.86236360303133458375920097279, 6.75883960639560868450780085036, 7.20928811675510186860511007909, 8.039235058620029408235004595080
