| L(s) = 1 | + 2-s + 0.584·3-s + 4-s + 0.414·5-s + 0.584·6-s + 7-s + 8-s − 2.65·9-s + 0.414·10-s − 3.50·11-s + 0.584·12-s − 3.47·13-s + 14-s + 0.242·15-s + 16-s − 1.24·17-s − 2.65·18-s + 6.13·19-s + 0.414·20-s + 0.584·21-s − 3.50·22-s + 0.633·23-s + 0.584·24-s − 4.82·25-s − 3.47·26-s − 3.30·27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.337·3-s + 0.5·4-s + 0.185·5-s + 0.238·6-s + 0.377·7-s + 0.353·8-s − 0.886·9-s + 0.131·10-s − 1.05·11-s + 0.168·12-s − 0.962·13-s + 0.267·14-s + 0.0625·15-s + 0.250·16-s − 0.302·17-s − 0.626·18-s + 1.40·19-s + 0.0927·20-s + 0.127·21-s − 0.746·22-s + 0.131·23-s + 0.119·24-s − 0.965·25-s − 0.680·26-s − 0.636·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 431 | \( 1 + T \) |
| good | 3 | \( 1 - 0.584T + 3T^{2} \) |
| 5 | \( 1 - 0.414T + 5T^{2} \) |
| 11 | \( 1 + 3.50T + 11T^{2} \) |
| 13 | \( 1 + 3.47T + 13T^{2} \) |
| 17 | \( 1 + 1.24T + 17T^{2} \) |
| 19 | \( 1 - 6.13T + 19T^{2} \) |
| 23 | \( 1 - 0.633T + 23T^{2} \) |
| 29 | \( 1 - 0.497T + 29T^{2} \) |
| 31 | \( 1 + 2.59T + 31T^{2} \) |
| 37 | \( 1 + 4.94T + 37T^{2} \) |
| 41 | \( 1 - 4.47T + 41T^{2} \) |
| 43 | \( 1 - 3.19T + 43T^{2} \) |
| 47 | \( 1 + 3.87T + 47T^{2} \) |
| 53 | \( 1 - 7.33T + 53T^{2} \) |
| 59 | \( 1 + 7.65T + 59T^{2} \) |
| 61 | \( 1 + 6.04T + 61T^{2} \) |
| 67 | \( 1 + 4.27T + 67T^{2} \) |
| 71 | \( 1 + 5.18T + 71T^{2} \) |
| 73 | \( 1 + 14.1T + 73T^{2} \) |
| 79 | \( 1 + 6.41T + 79T^{2} \) |
| 83 | \( 1 + 7.90T + 83T^{2} \) |
| 89 | \( 1 + 5.50T + 89T^{2} \) |
| 97 | \( 1 + 1.64T + 97T^{2} \) |
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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
−7.53290153393478288590349420374, −7.25169944546332240123990454108, −6.01070272787027837853162880464, −5.49709625997133488518692964619, −4.96027082767043789171421896304, −4.08036283990473024018889870261, −3.02024276879340156352252224685, −2.62838846431905808236448623454, −1.64545687297880235444528208379, 0,
1.64545687297880235444528208379, 2.62838846431905808236448623454, 3.02024276879340156352252224685, 4.08036283990473024018889870261, 4.96027082767043789171421896304, 5.49709625997133488518692964619, 6.01070272787027837853162880464, 7.25169944546332240123990454108, 7.53290153393478288590349420374
