| L(s) = 1 | + 1.88·2-s + 3-s + 1.53·4-s − 1.94·5-s + 1.88·6-s + 3.36·7-s − 0.865·8-s + 9-s − 3.66·10-s − 2.46·11-s + 1.53·12-s + 1.91·13-s + 6.32·14-s − 1.94·15-s − 4.70·16-s + 17-s + 1.88·18-s − 4.54·19-s − 2.99·20-s + 3.36·21-s − 4.64·22-s − 4.47·23-s − 0.865·24-s − 1.21·25-s + 3.59·26-s + 27-s + 5.17·28-s + ⋯ |
| L(s) = 1 | + 1.33·2-s + 0.577·3-s + 0.769·4-s − 0.870·5-s + 0.768·6-s + 1.27·7-s − 0.306·8-s + 0.333·9-s − 1.15·10-s − 0.744·11-s + 0.444·12-s + 0.529·13-s + 1.69·14-s − 0.502·15-s − 1.17·16-s + 0.242·17-s + 0.443·18-s − 1.04·19-s − 0.670·20-s + 0.733·21-s − 0.990·22-s − 0.932·23-s − 0.176·24-s − 0.242·25-s + 0.705·26-s + 0.192·27-s + 0.978·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 157 | \( 1 + T \) |
| good | 2 | \( 1 - 1.88T + 2T^{2} \) |
| 5 | \( 1 + 1.94T + 5T^{2} \) |
| 7 | \( 1 - 3.36T + 7T^{2} \) |
| 11 | \( 1 + 2.46T + 11T^{2} \) |
| 13 | \( 1 - 1.91T + 13T^{2} \) |
| 19 | \( 1 + 4.54T + 19T^{2} \) |
| 23 | \( 1 + 4.47T + 23T^{2} \) |
| 29 | \( 1 + 5.99T + 29T^{2} \) |
| 31 | \( 1 - 0.479T + 31T^{2} \) |
| 37 | \( 1 + 2.13T + 37T^{2} \) |
| 41 | \( 1 + 4.36T + 41T^{2} \) |
| 43 | \( 1 - 7.31T + 43T^{2} \) |
| 47 | \( 1 + 11.7T + 47T^{2} \) |
| 53 | \( 1 + 3.38T + 53T^{2} \) |
| 59 | \( 1 - 4.06T + 59T^{2} \) |
| 61 | \( 1 - 2.70T + 61T^{2} \) |
| 67 | \( 1 - 0.895T + 67T^{2} \) |
| 71 | \( 1 + 2.00T + 71T^{2} \) |
| 73 | \( 1 - 9.02T + 73T^{2} \) |
| 79 | \( 1 + 1.64T + 79T^{2} \) |
| 83 | \( 1 + 9.11T + 83T^{2} \) |
| 89 | \( 1 - 4.77T + 89T^{2} \) |
| 97 | \( 1 + 15.3T + 97T^{2} \) |
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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
−7.57383461380092767320702942660, −6.72746036620193665252956616768, −5.86585860948715883908314873043, −5.20778973573785368327066863747, −4.51902432942204531467730189622, −3.96964816703106169534232869594, −3.42691399753304148319746666446, −2.42336328833454803476623813886, −1.69550354046109916501928245744, 0,
1.69550354046109916501928245744, 2.42336328833454803476623813886, 3.42691399753304148319746666446, 3.96964816703106169534232869594, 4.51902432942204531467730189622, 5.20778973573785368327066863747, 5.86585860948715883908314873043, 6.72746036620193665252956616768, 7.57383461380092767320702942660
