L(s) = 1 | + 2-s − 0.0694·3-s + 4-s − 5-s − 0.0694·6-s + 0.700·7-s + 8-s − 2.99·9-s − 10-s − 1.18·11-s − 0.0694·12-s + 1.91·13-s + 0.700·14-s + 0.0694·15-s + 16-s + 3.77·17-s − 2.99·18-s − 2.42·19-s − 20-s − 0.0486·21-s − 1.18·22-s − 7.08·23-s − 0.0694·24-s + 25-s + 1.91·26-s + 0.416·27-s + 0.700·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.0400·3-s + 0.5·4-s − 0.447·5-s − 0.0283·6-s + 0.264·7-s + 0.353·8-s − 0.998·9-s − 0.316·10-s − 0.356·11-s − 0.0200·12-s + 0.531·13-s + 0.187·14-s + 0.0179·15-s + 0.250·16-s + 0.914·17-s − 0.705·18-s − 0.556·19-s − 0.223·20-s − 0.0106·21-s − 0.251·22-s − 1.47·23-s − 0.0141·24-s + 0.200·25-s + 0.375·26-s + 0.0801·27-s + 0.132·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 401 | \( 1 - T \) |
good | 3 | \( 1 + 0.0694T + 3T^{2} \) |
| 7 | \( 1 - 0.700T + 7T^{2} \) |
| 11 | \( 1 + 1.18T + 11T^{2} \) |
| 13 | \( 1 - 1.91T + 13T^{2} \) |
| 17 | \( 1 - 3.77T + 17T^{2} \) |
| 19 | \( 1 + 2.42T + 19T^{2} \) |
| 23 | \( 1 + 7.08T + 23T^{2} \) |
| 29 | \( 1 + 6.01T + 29T^{2} \) |
| 31 | \( 1 + 3.17T + 31T^{2} \) |
| 37 | \( 1 - 3.26T + 37T^{2} \) |
| 41 | \( 1 + 6.04T + 41T^{2} \) |
| 43 | \( 1 - 5.19T + 43T^{2} \) |
| 47 | \( 1 - 9.99T + 47T^{2} \) |
| 53 | \( 1 + 3.31T + 53T^{2} \) |
| 59 | \( 1 + 9.95T + 59T^{2} \) |
| 61 | \( 1 + 12.5T + 61T^{2} \) |
| 67 | \( 1 + 5.75T + 67T^{2} \) |
| 71 | \( 1 - 9.89T + 71T^{2} \) |
| 73 | \( 1 + 4.97T + 73T^{2} \) |
| 79 | \( 1 + 9.82T + 79T^{2} \) |
| 83 | \( 1 - 0.190T + 83T^{2} \) |
| 89 | \( 1 - 11.5T + 89T^{2} \) |
| 97 | \( 1 + 8.24T + 97T^{2} \) |
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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
−7.86623893707317148130932108876, −7.54887935754039196406386886420, −6.30742879094582366288473981013, −5.84915179886353588008084131321, −5.13884736281381453233714875877, −4.18275310326142754281217169680, −3.51840492527988846790104842782, −2.66671076205377683897196060629, −1.61258298493836190456644654992, 0,
1.61258298493836190456644654992, 2.66671076205377683897196060629, 3.51840492527988846790104842782, 4.18275310326142754281217169680, 5.13884736281381453233714875877, 5.84915179886353588008084131321, 6.30742879094582366288473981013, 7.54887935754039196406386886420, 7.86623893707317148130932108876