| L(s) = 1 | + 2-s − 1.74·3-s + 4-s + 2.80·5-s − 1.74·6-s − 7-s + 8-s + 0.0594·9-s + 2.80·10-s + 0.585·11-s − 1.74·12-s + 1.14·13-s − 14-s − 4.91·15-s + 16-s − 0.808·17-s + 0.0594·18-s − 1.49·19-s + 2.80·20-s + 1.74·21-s + 0.585·22-s − 1.74·24-s + 2.88·25-s + 1.14·26-s + 5.14·27-s − 28-s − 3.21·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.00·3-s + 0.5·4-s + 1.25·5-s − 0.714·6-s − 0.377·7-s + 0.353·8-s + 0.0198·9-s + 0.888·10-s + 0.176·11-s − 0.504·12-s + 0.317·13-s − 0.267·14-s − 1.26·15-s + 0.250·16-s − 0.196·17-s + 0.0140·18-s − 0.343·19-s + 0.628·20-s + 0.381·21-s + 0.124·22-s − 0.357·24-s + 0.577·25-s + 0.224·26-s + 0.989·27-s − 0.188·28-s − 0.597·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7406 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7406 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.644669333\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.644669333\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + 1.74T + 3T^{2} \) |
| 5 | \( 1 - 2.80T + 5T^{2} \) |
| 11 | \( 1 - 0.585T + 11T^{2} \) |
| 13 | \( 1 - 1.14T + 13T^{2} \) |
| 17 | \( 1 + 0.808T + 17T^{2} \) |
| 19 | \( 1 + 1.49T + 19T^{2} \) |
| 29 | \( 1 + 3.21T + 29T^{2} \) |
| 31 | \( 1 + 1.86T + 31T^{2} \) |
| 37 | \( 1 - 2.36T + 37T^{2} \) |
| 41 | \( 1 - 0.550T + 41T^{2} \) |
| 43 | \( 1 - 12.6T + 43T^{2} \) |
| 47 | \( 1 - 6.85T + 47T^{2} \) |
| 53 | \( 1 + 9.19T + 53T^{2} \) |
| 59 | \( 1 - 3.86T + 59T^{2} \) |
| 61 | \( 1 + 7.96T + 61T^{2} \) |
| 67 | \( 1 - 12.3T + 67T^{2} \) |
| 71 | \( 1 - 11.2T + 71T^{2} \) |
| 73 | \( 1 - 6.20T + 73T^{2} \) |
| 79 | \( 1 + 13.2T + 79T^{2} \) |
| 83 | \( 1 + 2.85T + 83T^{2} \) |
| 89 | \( 1 - 12.0T + 89T^{2} \) |
| 97 | \( 1 + 0.955T + 97T^{2} \) |
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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
−7.64783092086435698353157418456, −6.84114954179124962520652318205, −6.14850010470286740486021584237, −5.91491486260152061335413109719, −5.28010713135878994129075359876, −4.51969501418063769287504193870, −3.64918563061939980492914108400, −2.64625881869803142309015420814, −1.91281510791106754367079275469, −0.77069789455870825583072016594,
0.77069789455870825583072016594, 1.91281510791106754367079275469, 2.64625881869803142309015420814, 3.64918563061939980492914108400, 4.51969501418063769287504193870, 5.28010713135878994129075359876, 5.91491486260152061335413109719, 6.14850010470286740486021584237, 6.84114954179124962520652318205, 7.64783092086435698353157418456
