L(s) = 1 | + 2-s − 0.618·3-s + 4-s + 5-s − 0.618·6-s − 7-s + 8-s − 2.61·9-s + 10-s − 5·11-s − 0.618·12-s + 4.85·13-s − 14-s − 0.618·15-s + 16-s + 6.85·17-s − 2.61·18-s + 20-s + 0.618·21-s − 5·22-s − 3.76·23-s − 0.618·24-s − 4·25-s + 4.85·26-s + 3.47·27-s − 28-s − 8.23·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.356·3-s + 0.5·4-s + 0.447·5-s − 0.252·6-s − 0.377·7-s + 0.353·8-s − 0.872·9-s + 0.316·10-s − 1.50·11-s − 0.178·12-s + 1.34·13-s − 0.267·14-s − 0.159·15-s + 0.250·16-s + 1.66·17-s − 0.617·18-s + 0.223·20-s + 0.134·21-s − 1.06·22-s − 0.784·23-s − 0.126·24-s − 0.800·25-s + 0.951·26-s + 0.668·27-s − 0.188·28-s − 1.52·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + 0.618T + 3T^{2} \) |
| 5 | \( 1 - T + 5T^{2} \) |
| 11 | \( 1 + 5T + 11T^{2} \) |
| 13 | \( 1 - 4.85T + 13T^{2} \) |
| 17 | \( 1 - 6.85T + 17T^{2} \) |
| 23 | \( 1 + 3.76T + 23T^{2} \) |
| 29 | \( 1 + 8.23T + 29T^{2} \) |
| 31 | \( 1 + 6.70T + 31T^{2} \) |
| 37 | \( 1 - 5.94T + 37T^{2} \) |
| 41 | \( 1 + 1.09T + 41T^{2} \) |
| 43 | \( 1 + 6.85T + 43T^{2} \) |
| 47 | \( 1 - 7.94T + 47T^{2} \) |
| 53 | \( 1 + 12.7T + 53T^{2} \) |
| 59 | \( 1 - 11T + 59T^{2} \) |
| 61 | \( 1 - 7.47T + 61T^{2} \) |
| 67 | \( 1 - 7.85T + 67T^{2} \) |
| 71 | \( 1 + 10T + 71T^{2} \) |
| 73 | \( 1 + 14.5T + 73T^{2} \) |
| 79 | \( 1 + 8.85T + 79T^{2} \) |
| 83 | \( 1 + 15.4T + 83T^{2} \) |
| 89 | \( 1 - 4.85T + 89T^{2} \) |
| 97 | \( 1 + 9.47T + 97T^{2} \) |
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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
−7.88715610108539321239776415825, −7.06845957627723859894693942724, −5.96780323568502429672394527833, −5.67776377120205288220024769371, −5.34975056671965840447134039218, −4.02620808779089183486358775718, −3.34746484326330225806900423569, −2.58307551410786850409947326964, −1.52007451308014621548898381245, 0,
1.52007451308014621548898381245, 2.58307551410786850409947326964, 3.34746484326330225806900423569, 4.02620808779089183486358775718, 5.34975056671965840447134039218, 5.67776377120205288220024769371, 5.96780323568502429672394527833, 7.06845957627723859894693942724, 7.88715610108539321239776415825