| L(s) = 1 | + 2.67·2-s − 3-s + 5.15·4-s − 2.95·5-s − 2.67·6-s − 0.0201·7-s + 8.42·8-s + 9-s − 7.89·10-s − 5.62·11-s − 5.15·12-s + 13-s − 0.0537·14-s + 2.95·15-s + 12.2·16-s − 0.758·17-s + 2.67·18-s + 3.15·19-s − 15.1·20-s + 0.0201·21-s − 15.0·22-s − 4.90·23-s − 8.42·24-s + 3.70·25-s + 2.67·26-s − 27-s − 0.103·28-s + ⋯ |
| L(s) = 1 | + 1.89·2-s − 0.577·3-s + 2.57·4-s − 1.31·5-s − 1.09·6-s − 0.00760·7-s + 2.97·8-s + 0.333·9-s − 2.49·10-s − 1.69·11-s − 1.48·12-s + 0.277·13-s − 0.0143·14-s + 0.761·15-s + 3.05·16-s − 0.183·17-s + 0.630·18-s + 0.723·19-s − 3.39·20-s + 0.00438·21-s − 3.20·22-s − 1.02·23-s − 1.71·24-s + 0.741·25-s + 0.524·26-s − 0.192·27-s − 0.0195·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
| good | 2 | \( 1 - 2.67T + 2T^{2} \) |
| 5 | \( 1 + 2.95T + 5T^{2} \) |
| 7 | \( 1 + 0.0201T + 7T^{2} \) |
| 11 | \( 1 + 5.62T + 11T^{2} \) |
| 17 | \( 1 + 0.758T + 17T^{2} \) |
| 19 | \( 1 - 3.15T + 19T^{2} \) |
| 23 | \( 1 + 4.90T + 23T^{2} \) |
| 29 | \( 1 + 2.21T + 29T^{2} \) |
| 31 | \( 1 + 4.29T + 31T^{2} \) |
| 37 | \( 1 + 1.23T + 37T^{2} \) |
| 41 | \( 1 + 4.18T + 41T^{2} \) |
| 43 | \( 1 + 5.20T + 43T^{2} \) |
| 47 | \( 1 - 9.12T + 47T^{2} \) |
| 53 | \( 1 + 12.3T + 53T^{2} \) |
| 59 | \( 1 + 8.26T + 59T^{2} \) |
| 61 | \( 1 - 5.46T + 61T^{2} \) |
| 67 | \( 1 - 4.32T + 67T^{2} \) |
| 71 | \( 1 + 10.2T + 71T^{2} \) |
| 73 | \( 1 + 7.80T + 73T^{2} \) |
| 79 | \( 1 + 0.124T + 79T^{2} \) |
| 83 | \( 1 - 4.62T + 83T^{2} \) |
| 89 | \( 1 + 4.27T + 89T^{2} \) |
| 97 | \( 1 + 2.84T + 97T^{2} \) |
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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
−7.61723790318471075974013639583, −7.32301715395475492133409578467, −6.33605616026110211644964866594, −5.60456136853896142583810864689, −5.01385087099443587692383930520, −4.36304783561940680218992840278, −3.57893144560498522212408711461, −2.95902315809307364659447629065, −1.81831284408673235989654853222, 0,
1.81831284408673235989654853222, 2.95902315809307364659447629065, 3.57893144560498522212408711461, 4.36304783561940680218992840278, 5.01385087099443587692383930520, 5.60456136853896142583810864689, 6.33605616026110211644964866594, 7.32301715395475492133409578467, 7.61723790318471075974013639583
