| L(s) = 1 | − 2-s − 0.666·3-s + 4-s + 0.513·5-s + 0.666·6-s + 2.90·7-s − 8-s − 2.55·9-s − 0.513·10-s + 0.858·11-s − 0.666·12-s − 5.16·13-s − 2.90·14-s − 0.342·15-s + 16-s + 4.36·17-s + 2.55·18-s − 1.53·19-s + 0.513·20-s − 1.93·21-s − 0.858·22-s − 23-s + 0.666·24-s − 4.73·25-s + 5.16·26-s + 3.70·27-s + 2.90·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.384·3-s + 0.5·4-s + 0.229·5-s + 0.271·6-s + 1.09·7-s − 0.353·8-s − 0.852·9-s − 0.162·10-s + 0.258·11-s − 0.192·12-s − 1.43·13-s − 0.775·14-s − 0.0883·15-s + 0.250·16-s + 1.05·17-s + 0.602·18-s − 0.353·19-s + 0.114·20-s − 0.421·21-s − 0.183·22-s − 0.208·23-s + 0.135·24-s − 0.947·25-s + 1.01·26-s + 0.712·27-s + 0.548·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 + T \) |
| good | 3 | \( 1 + 0.666T + 3T^{2} \) |
| 5 | \( 1 - 0.513T + 5T^{2} \) |
| 7 | \( 1 - 2.90T + 7T^{2} \) |
| 11 | \( 1 - 0.858T + 11T^{2} \) |
| 13 | \( 1 + 5.16T + 13T^{2} \) |
| 17 | \( 1 - 4.36T + 17T^{2} \) |
| 19 | \( 1 + 1.53T + 19T^{2} \) |
| 29 | \( 1 + 3.13T + 29T^{2} \) |
| 31 | \( 1 - 9.58T + 31T^{2} \) |
| 37 | \( 1 + 7.80T + 37T^{2} \) |
| 41 | \( 1 - 4.19T + 41T^{2} \) |
| 43 | \( 1 - 11.0T + 43T^{2} \) |
| 47 | \( 1 + 1.61T + 47T^{2} \) |
| 53 | \( 1 + 6.05T + 53T^{2} \) |
| 59 | \( 1 - 3.21T + 59T^{2} \) |
| 61 | \( 1 + 6.32T + 61T^{2} \) |
| 67 | \( 1 + 3.00T + 67T^{2} \) |
| 71 | \( 1 - 2.11T + 71T^{2} \) |
| 73 | \( 1 - 12.7T + 73T^{2} \) |
| 79 | \( 1 + 2.80T + 79T^{2} \) |
| 83 | \( 1 + 16.3T + 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 + 13.5T + 97T^{2} \) |
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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
−7.85911210833871846361089662055, −7.20832148074088115866585424337, −6.28460021773355756224146114606, −5.60532910709166266485315332877, −5.02170511145157460407069769366, −4.15267705898469082568963335144, −2.92677988388254805051053829658, −2.19607620679249243580155242508, −1.23180008523943881369870375559, 0,
1.23180008523943881369870375559, 2.19607620679249243580155242508, 2.92677988388254805051053829658, 4.15267705898469082568963335144, 5.02170511145157460407069769366, 5.60532910709166266485315332877, 6.28460021773355756224146114606, 7.20832148074088115866585424337, 7.85911210833871846361089662055
