| L(s) = 1 | + 2-s + 4-s + 5-s + 3.86·7-s + 8-s + 10-s − 6.49·11-s − 5.66·13-s + 3.86·14-s + 16-s − 0.442·17-s − 6.08·19-s + 20-s − 6.49·22-s − 0.182·23-s + 25-s − 5.66·26-s + 3.86·28-s − 6.73·29-s + 8.06·31-s + 32-s − 0.442·34-s + 3.86·35-s − 0.643·37-s − 6.08·38-s + 40-s − 10.5·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.447·5-s + 1.46·7-s + 0.353·8-s + 0.316·10-s − 1.95·11-s − 1.57·13-s + 1.03·14-s + 0.250·16-s − 0.107·17-s − 1.39·19-s + 0.223·20-s − 1.38·22-s − 0.0379·23-s + 0.200·25-s − 1.11·26-s + 0.730·28-s − 1.25·29-s + 1.44·31-s + 0.176·32-s − 0.0759·34-s + 0.653·35-s − 0.105·37-s − 0.986·38-s + 0.158·40-s − 1.64·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 67 | \( 1 - T \) |
| good | 7 | \( 1 - 3.86T + 7T^{2} \) |
| 11 | \( 1 + 6.49T + 11T^{2} \) |
| 13 | \( 1 + 5.66T + 13T^{2} \) |
| 17 | \( 1 + 0.442T + 17T^{2} \) |
| 19 | \( 1 + 6.08T + 19T^{2} \) |
| 23 | \( 1 + 0.182T + 23T^{2} \) |
| 29 | \( 1 + 6.73T + 29T^{2} \) |
| 31 | \( 1 - 8.06T + 31T^{2} \) |
| 37 | \( 1 + 0.643T + 37T^{2} \) |
| 41 | \( 1 + 10.5T + 41T^{2} \) |
| 43 | \( 1 - 1.77T + 43T^{2} \) |
| 47 | \( 1 - 8.64T + 47T^{2} \) |
| 53 | \( 1 + 11.7T + 53T^{2} \) |
| 59 | \( 1 + 8.09T + 59T^{2} \) |
| 61 | \( 1 + 8.33T + 61T^{2} \) |
| 71 | \( 1 + 15.8T + 71T^{2} \) |
| 73 | \( 1 - 3.19T + 73T^{2} \) |
| 79 | \( 1 - 2.08T + 79T^{2} \) |
| 83 | \( 1 + 1.63T + 83T^{2} \) |
| 89 | \( 1 + 8.49T + 89T^{2} \) |
| 97 | \( 1 - 1.59T + 97T^{2} \) |
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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
−7.72983663523975834886714690319, −7.11317266123431367223861544582, −6.07820188875346343551765704891, −5.39567178590668755462113157103, −4.72030185253288735008404869598, −4.53403320190455318121619761191, −3.04061380106911955873032280787, −2.31820173610210254847794851475, −1.78133942098203999076723859014, 0,
1.78133942098203999076723859014, 2.31820173610210254847794851475, 3.04061380106911955873032280787, 4.53403320190455318121619761191, 4.72030185253288735008404869598, 5.39567178590668755462113157103, 6.07820188875346343551765704891, 7.11317266123431367223861544582, 7.72983663523975834886714690319
