| L(s) = 1 | + 2-s − 0.492·3-s + 4-s − 0.547·5-s − 0.492·6-s + 4.87·7-s + 8-s − 2.75·9-s − 0.547·10-s − 1.69·11-s − 0.492·12-s − 3.38·13-s + 4.87·14-s + 0.269·15-s + 16-s − 3.46·17-s − 2.75·18-s − 3.29·19-s − 0.547·20-s − 2.40·21-s − 1.69·22-s + 6.12·23-s − 0.492·24-s − 4.70·25-s − 3.38·26-s + 2.83·27-s + 4.87·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.284·3-s + 0.5·4-s − 0.244·5-s − 0.201·6-s + 1.84·7-s + 0.353·8-s − 0.919·9-s − 0.173·10-s − 0.510·11-s − 0.142·12-s − 0.938·13-s + 1.30·14-s + 0.0695·15-s + 0.250·16-s − 0.841·17-s − 0.649·18-s − 0.755·19-s − 0.122·20-s − 0.524·21-s − 0.360·22-s + 1.27·23-s − 0.100·24-s − 0.940·25-s − 0.663·26-s + 0.545·27-s + 0.922·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.833480724\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.833480724\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| 97 | \( 1 + T \) |
| good | 3 | \( 1 + 0.492T + 3T^{2} \) |
| 5 | \( 1 + 0.547T + 5T^{2} \) |
| 7 | \( 1 - 4.87T + 7T^{2} \) |
| 11 | \( 1 + 1.69T + 11T^{2} \) |
| 13 | \( 1 + 3.38T + 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 + 3.29T + 19T^{2} \) |
| 23 | \( 1 - 6.12T + 23T^{2} \) |
| 29 | \( 1 - 3.87T + 29T^{2} \) |
| 37 | \( 1 - 3.70T + 37T^{2} \) |
| 41 | \( 1 - 4.48T + 41T^{2} \) |
| 43 | \( 1 - 5.13T + 43T^{2} \) |
| 47 | \( 1 - 1.82T + 47T^{2} \) |
| 53 | \( 1 - 12.1T + 53T^{2} \) |
| 59 | \( 1 - 2.18T + 59T^{2} \) |
| 61 | \( 1 - 9.59T + 61T^{2} \) |
| 67 | \( 1 - 13.3T + 67T^{2} \) |
| 71 | \( 1 - 6.24T + 71T^{2} \) |
| 73 | \( 1 - 6.47T + 73T^{2} \) |
| 79 | \( 1 + 4.26T + 79T^{2} \) |
| 83 | \( 1 + 3.67T + 83T^{2} \) |
| 89 | \( 1 - 14.3T + 89T^{2} \) |
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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
−8.055364276609104283510163325056, −7.36133564544476188960049977671, −6.64207821666587220606987834322, −5.62857727992906121590496762210, −5.17212647347118111407992847529, −4.58820961533858668890441128399, −3.91151848636474914386800602889, −2.50667002384936679061040240801, −2.25535025894868238159694834332, −0.794156719716061280376574424736,
0.794156719716061280376574424736, 2.25535025894868238159694834332, 2.50667002384936679061040240801, 3.91151848636474914386800602889, 4.58820961533858668890441128399, 5.17212647347118111407992847529, 5.62857727992906121590496762210, 6.64207821666587220606987834322, 7.36133564544476188960049977671, 8.055364276609104283510163325056
