| L(s) = 1 | + 2-s + 4-s − 3.27·5-s + 4.39·7-s + 8-s − 3.27·10-s − 2.69·11-s + 3.54·13-s + 4.39·14-s + 16-s − 2.86·17-s − 3.06·19-s − 3.27·20-s − 2.69·22-s + 4.38·23-s + 5.73·25-s + 3.54·26-s + 4.39·28-s − 10.5·29-s − 9.78·31-s + 32-s − 2.86·34-s − 14.4·35-s + 2.90·37-s − 3.06·38-s − 3.27·40-s + 1.53·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.46·5-s + 1.66·7-s + 0.353·8-s − 1.03·10-s − 0.812·11-s + 0.984·13-s + 1.17·14-s + 0.250·16-s − 0.695·17-s − 0.703·19-s − 0.732·20-s − 0.574·22-s + 0.915·23-s + 1.14·25-s + 0.695·26-s + 0.831·28-s − 1.95·29-s − 1.75·31-s + 0.176·32-s − 0.491·34-s − 2.43·35-s + 0.477·37-s − 0.497·38-s − 0.517·40-s + 0.239·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 149 | \( 1 - T \) |
| good | 5 | \( 1 + 3.27T + 5T^{2} \) |
| 7 | \( 1 - 4.39T + 7T^{2} \) |
| 11 | \( 1 + 2.69T + 11T^{2} \) |
| 13 | \( 1 - 3.54T + 13T^{2} \) |
| 17 | \( 1 + 2.86T + 17T^{2} \) |
| 19 | \( 1 + 3.06T + 19T^{2} \) |
| 23 | \( 1 - 4.38T + 23T^{2} \) |
| 29 | \( 1 + 10.5T + 29T^{2} \) |
| 31 | \( 1 + 9.78T + 31T^{2} \) |
| 37 | \( 1 - 2.90T + 37T^{2} \) |
| 41 | \( 1 - 1.53T + 41T^{2} \) |
| 43 | \( 1 + 9.57T + 43T^{2} \) |
| 47 | \( 1 - 4.85T + 47T^{2} \) |
| 53 | \( 1 + 2.73T + 53T^{2} \) |
| 59 | \( 1 + 8.65T + 59T^{2} \) |
| 61 | \( 1 - 0.791T + 61T^{2} \) |
| 67 | \( 1 + 4.78T + 67T^{2} \) |
| 71 | \( 1 - 9.73T + 71T^{2} \) |
| 73 | \( 1 - 9.30T + 73T^{2} \) |
| 79 | \( 1 + 10.1T + 79T^{2} \) |
| 83 | \( 1 + 5.84T + 83T^{2} \) |
| 89 | \( 1 + 15.2T + 89T^{2} \) |
| 97 | \( 1 - 16.9T + 97T^{2} \) |
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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
−7.48745603696485361596969512744, −7.01039391277946210574467081666, −5.90141421279204413345783219435, −5.20477373955694467974200403171, −4.61261597595089859956606025949, −3.97158710212868245260249721204, −3.40560695303989898290950654334, −2.24967063987078783656738625418, −1.44140051336084528774294600209, 0,
1.44140051336084528774294600209, 2.24967063987078783656738625418, 3.40560695303989898290950654334, 3.97158710212868245260249721204, 4.61261597595089859956606025949, 5.20477373955694467974200403171, 5.90141421279204413345783219435, 7.01039391277946210574467081666, 7.48745603696485361596969512744
