| L(s) = 1 | + 1.79·2-s − 3-s + 1.20·4-s + 5-s − 1.79·6-s − 4.79·7-s − 1.41·8-s + 9-s + 1.79·10-s − 6.14·11-s − 1.20·12-s − 1.49·13-s − 8.59·14-s − 15-s − 4.95·16-s − 5.08·17-s + 1.79·18-s − 2.83·19-s + 1.20·20-s + 4.79·21-s − 11.0·22-s + 1.41·24-s + 25-s − 2.66·26-s − 27-s − 5.79·28-s + 5.95·29-s + ⋯ |
| L(s) = 1 | + 1.26·2-s − 0.577·3-s + 0.603·4-s + 0.447·5-s − 0.731·6-s − 1.81·7-s − 0.501·8-s + 0.333·9-s + 0.566·10-s − 1.85·11-s − 0.348·12-s − 0.413·13-s − 2.29·14-s − 0.258·15-s − 1.23·16-s − 1.23·17-s + 0.422·18-s − 0.650·19-s + 0.270·20-s + 1.04·21-s − 2.34·22-s + 0.289·24-s + 0.200·25-s − 0.523·26-s − 0.192·27-s − 1.09·28-s + 1.10·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7106553595\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7106553595\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 - 1.79T + 2T^{2} \) |
| 7 | \( 1 + 4.79T + 7T^{2} \) |
| 11 | \( 1 + 6.14T + 11T^{2} \) |
| 13 | \( 1 + 1.49T + 13T^{2} \) |
| 17 | \( 1 + 5.08T + 17T^{2} \) |
| 19 | \( 1 + 2.83T + 19T^{2} \) |
| 29 | \( 1 - 5.95T + 29T^{2} \) |
| 31 | \( 1 + 7.06T + 31T^{2} \) |
| 37 | \( 1 + 1.77T + 37T^{2} \) |
| 41 | \( 1 - 4.42T + 41T^{2} \) |
| 43 | \( 1 + 0.0523T + 43T^{2} \) |
| 47 | \( 1 - 9.00T + 47T^{2} \) |
| 53 | \( 1 - 12.1T + 53T^{2} \) |
| 59 | \( 1 - 1.48T + 59T^{2} \) |
| 61 | \( 1 + 2.61T + 61T^{2} \) |
| 67 | \( 1 - 2.39T + 67T^{2} \) |
| 71 | \( 1 + 14.8T + 71T^{2} \) |
| 73 | \( 1 + 3.46T + 73T^{2} \) |
| 79 | \( 1 - 1.41T + 79T^{2} \) |
| 83 | \( 1 - 9.77T + 83T^{2} \) |
| 89 | \( 1 - 6.02T + 89T^{2} \) |
| 97 | \( 1 + 0.574T + 97T^{2} \) |
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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
−7.42411340568852260105939957157, −6.85689663872322514084855913933, −6.21246388517229723013214274344, −5.71286749164997364044853658705, −5.13003910365324013973670425129, −4.37677022678986571060044270209, −3.64235927612761280312423958009, −2.60534622465442686134373600112, −2.45584259395718258229361370112, −0.32047360890981492943705407285,
0.32047360890981492943705407285, 2.45584259395718258229361370112, 2.60534622465442686134373600112, 3.64235927612761280312423958009, 4.37677022678986571060044270209, 5.13003910365324013973670425129, 5.71286749164997364044853658705, 6.21246388517229723013214274344, 6.85689663872322514084855913933, 7.42411340568852260105939957157
