| L(s) = 1 | + 2-s − 0.197·3-s + 4-s − 0.308·5-s − 0.197·6-s + 8-s − 2.96·9-s − 0.308·10-s + 0.369·11-s − 0.197·12-s + 6.29·13-s + 0.0608·15-s + 16-s + 4.61·17-s − 2.96·18-s − 0.352·19-s − 0.308·20-s + 0.369·22-s − 23-s − 0.197·24-s − 4.90·25-s + 6.29·26-s + 1.17·27-s + 1.74·29-s + 0.0608·30-s + 2.93·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.113·3-s + 0.5·4-s − 0.138·5-s − 0.0804·6-s + 0.353·8-s − 0.987·9-s − 0.0976·10-s + 0.111·11-s − 0.0569·12-s + 1.74·13-s + 0.0157·15-s + 0.250·16-s + 1.11·17-s − 0.697·18-s − 0.0809·19-s − 0.0690·20-s + 0.0788·22-s − 0.208·23-s − 0.0402·24-s − 0.980·25-s + 1.23·26-s + 0.226·27-s + 0.323·29-s + 0.0111·30-s + 0.527·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2254 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2254 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.730372549\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.730372549\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + 0.197T + 3T^{2} \) |
| 5 | \( 1 + 0.308T + 5T^{2} \) |
| 11 | \( 1 - 0.369T + 11T^{2} \) |
| 13 | \( 1 - 6.29T + 13T^{2} \) |
| 17 | \( 1 - 4.61T + 17T^{2} \) |
| 19 | \( 1 + 0.352T + 19T^{2} \) |
| 29 | \( 1 - 1.74T + 29T^{2} \) |
| 31 | \( 1 - 2.93T + 31T^{2} \) |
| 37 | \( 1 + 6.37T + 37T^{2} \) |
| 41 | \( 1 + 1.90T + 41T^{2} \) |
| 43 | \( 1 - 6.55T + 43T^{2} \) |
| 47 | \( 1 - 11.7T + 47T^{2} \) |
| 53 | \( 1 - 5.90T + 53T^{2} \) |
| 59 | \( 1 + 2.29T + 59T^{2} \) |
| 61 | \( 1 - 6.78T + 61T^{2} \) |
| 67 | \( 1 - 12.7T + 67T^{2} \) |
| 71 | \( 1 - 0.492T + 71T^{2} \) |
| 73 | \( 1 - 5.99T + 73T^{2} \) |
| 79 | \( 1 - 0.666T + 79T^{2} \) |
| 83 | \( 1 + 6.44T + 83T^{2} \) |
| 89 | \( 1 - 11.6T + 89T^{2} \) |
| 97 | \( 1 - 4.96T + 97T^{2} \) |
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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
−8.815974934810392366497812874286, −8.287805558565876203649843802551, −7.45447026366864740010998596393, −6.40048006897979604213590806826, −5.84497241777721103434756008513, −5.22591726869216901375151198657, −3.94974088909161569979003744906, −3.48143524764500583999337294426, −2.37113816147583115014284277847, −1.01620770014978187730962344575,
1.01620770014978187730962344575, 2.37113816147583115014284277847, 3.48143524764500583999337294426, 3.94974088909161569979003744906, 5.22591726869216901375151198657, 5.84497241777721103434756008513, 6.40048006897979604213590806826, 7.45447026366864740010998596393, 8.287805558565876203649843802551, 8.815974934810392366497812874286
