| L(s) = 1 | + 2-s + 4-s − 4.35·5-s + 3.05·7-s + 8-s − 4.35·10-s + 0.734·11-s − 2.43·13-s + 3.05·14-s + 16-s + 0.681·17-s − 1.87·19-s − 4.35·20-s + 0.734·22-s − 6.91·23-s + 13.9·25-s − 2.43·26-s + 3.05·28-s − 0.113·29-s + 1.56·31-s + 32-s + 0.681·34-s − 13.3·35-s − 3.71·37-s − 1.87·38-s − 4.35·40-s + 11.1·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.94·5-s + 1.15·7-s + 0.353·8-s − 1.37·10-s + 0.221·11-s − 0.675·13-s + 0.816·14-s + 0.250·16-s + 0.165·17-s − 0.430·19-s − 0.973·20-s + 0.156·22-s − 1.44·23-s + 2.79·25-s − 0.477·26-s + 0.577·28-s − 0.0210·29-s + 0.281·31-s + 0.176·32-s + 0.116·34-s − 2.25·35-s − 0.611·37-s − 0.304·38-s − 0.688·40-s + 1.73·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 223 | \( 1 + T \) |
| good | 5 | \( 1 + 4.35T + 5T^{2} \) |
| 7 | \( 1 - 3.05T + 7T^{2} \) |
| 11 | \( 1 - 0.734T + 11T^{2} \) |
| 13 | \( 1 + 2.43T + 13T^{2} \) |
| 17 | \( 1 - 0.681T + 17T^{2} \) |
| 19 | \( 1 + 1.87T + 19T^{2} \) |
| 23 | \( 1 + 6.91T + 23T^{2} \) |
| 29 | \( 1 + 0.113T + 29T^{2} \) |
| 31 | \( 1 - 1.56T + 31T^{2} \) |
| 37 | \( 1 + 3.71T + 37T^{2} \) |
| 41 | \( 1 - 11.1T + 41T^{2} \) |
| 43 | \( 1 + 10.9T + 43T^{2} \) |
| 47 | \( 1 - 6.22T + 47T^{2} \) |
| 53 | \( 1 + 6.47T + 53T^{2} \) |
| 59 | \( 1 + 7.94T + 59T^{2} \) |
| 61 | \( 1 + 4.54T + 61T^{2} \) |
| 67 | \( 1 + 2.23T + 67T^{2} \) |
| 71 | \( 1 + 10.5T + 71T^{2} \) |
| 73 | \( 1 + 3.30T + 73T^{2} \) |
| 79 | \( 1 + 3.98T + 79T^{2} \) |
| 83 | \( 1 - 9.77T + 83T^{2} \) |
| 89 | \( 1 - 3.23T + 89T^{2} \) |
| 97 | \( 1 + 11.1T + 97T^{2} \) |
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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
−8.006023430992368384880759557122, −7.45535136165714044705727743445, −6.74432647671267327720480468040, −5.70566330189309827559873563861, −4.69249384697693374410123973765, −4.39025823189463432230473701386, −3.65366665827196988532709149497, −2.70708682476567549062205826560, −1.49385049171834636435184798567, 0,
1.49385049171834636435184798567, 2.70708682476567549062205826560, 3.65366665827196988532709149497, 4.39025823189463432230473701386, 4.69249384697693374410123973765, 5.70566330189309827559873563861, 6.74432647671267327720480468040, 7.45535136165714044705727743445, 8.006023430992368384880759557122
