| L(s) = 1 | − 2-s − 3.41·3-s + 4-s − 5-s + 3.41·6-s − 8-s + 8.65·9-s + 10-s − 0.828·11-s − 3.41·12-s + 4.82·13-s + 3.41·15-s + 16-s − 2.58·17-s − 8.65·18-s − 0.585·19-s − 20-s + 0.828·22-s − 1.17·23-s + 3.41·24-s + 25-s − 4.82·26-s − 19.3·27-s − 4.82·29-s − 3.41·30-s + 2.82·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.97·3-s + 0.5·4-s − 0.447·5-s + 1.39·6-s − 0.353·8-s + 2.88·9-s + 0.316·10-s − 0.249·11-s − 0.985·12-s + 1.33·13-s + 0.881·15-s + 0.250·16-s − 0.627·17-s − 2.04·18-s − 0.134·19-s − 0.223·20-s + 0.176·22-s − 0.244·23-s + 0.696·24-s + 0.200·25-s − 0.946·26-s − 3.71·27-s − 0.896·29-s − 0.623·30-s + 0.508·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 3.41T + 3T^{2} \) |
| 11 | \( 1 + 0.828T + 11T^{2} \) |
| 13 | \( 1 - 4.82T + 13T^{2} \) |
| 17 | \( 1 + 2.58T + 17T^{2} \) |
| 19 | \( 1 + 0.585T + 19T^{2} \) |
| 23 | \( 1 + 1.17T + 23T^{2} \) |
| 29 | \( 1 + 4.82T + 29T^{2} \) |
| 31 | \( 1 - 2.82T + 31T^{2} \) |
| 37 | \( 1 + 7.65T + 37T^{2} \) |
| 41 | \( 1 - 3.07T + 41T^{2} \) |
| 43 | \( 1 + 8.82T + 43T^{2} \) |
| 47 | \( 1 + 5.17T + 47T^{2} \) |
| 53 | \( 1 - 6.48T + 53T^{2} \) |
| 59 | \( 1 + 8.58T + 59T^{2} \) |
| 61 | \( 1 + 9.31T + 61T^{2} \) |
| 67 | \( 1 - 1.65T + 67T^{2} \) |
| 71 | \( 1 + 4.48T + 71T^{2} \) |
| 73 | \( 1 - 9.41T + 73T^{2} \) |
| 79 | \( 1 + 6.82T + 79T^{2} \) |
| 83 | \( 1 - 2.24T + 83T^{2} \) |
| 89 | \( 1 + 12.7T + 89T^{2} \) |
| 97 | \( 1 - 7.75T + 97T^{2} \) |
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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
−10.73213762512098074802689265877, −9.955701569759992037364841241935, −8.762847715714262290796433712972, −7.62368940817791412410796986024, −6.68130388111122230056375947544, −6.02993191575522903379874152841, −4.99068908344961615257254006412, −3.82915150342024593183245811160, −1.48713854200108292571192289916, 0,
1.48713854200108292571192289916, 3.82915150342024593183245811160, 4.99068908344961615257254006412, 6.02993191575522903379874152841, 6.68130388111122230056375947544, 7.62368940817791412410796986024, 8.762847715714262290796433712972, 9.955701569759992037364841241935, 10.73213762512098074802689265877
