| L(s) = 1 | + 2.67·2-s + 5.15·4-s − 1.93·5-s + 0.721·7-s + 8.44·8-s − 5.17·10-s + 1.21·11-s + 0.759·13-s + 1.93·14-s + 12.2·16-s + 7.43·17-s + 0.975·19-s − 9.96·20-s + 3.25·22-s − 23-s − 1.26·25-s + 2.03·26-s + 3.72·28-s − 29-s + 4.89·31-s + 15.9·32-s + 19.8·34-s − 1.39·35-s − 7.11·37-s + 2.61·38-s − 16.3·40-s − 6.93·41-s + ⋯ |
| L(s) = 1 | + 1.89·2-s + 2.57·4-s − 0.864·5-s + 0.272·7-s + 2.98·8-s − 1.63·10-s + 0.367·11-s + 0.210·13-s + 0.516·14-s + 3.07·16-s + 1.80·17-s + 0.223·19-s − 2.22·20-s + 0.694·22-s − 0.208·23-s − 0.253·25-s + 0.398·26-s + 0.703·28-s − 0.185·29-s + 0.878·31-s + 2.82·32-s + 3.41·34-s − 0.235·35-s − 1.17·37-s + 0.423·38-s − 2.58·40-s − 1.08·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.117305711\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.117305711\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - 2.67T + 2T^{2} \) |
| 5 | \( 1 + 1.93T + 5T^{2} \) |
| 7 | \( 1 - 0.721T + 7T^{2} \) |
| 11 | \( 1 - 1.21T + 11T^{2} \) |
| 13 | \( 1 - 0.759T + 13T^{2} \) |
| 17 | \( 1 - 7.43T + 17T^{2} \) |
| 19 | \( 1 - 0.975T + 19T^{2} \) |
| 31 | \( 1 - 4.89T + 31T^{2} \) |
| 37 | \( 1 + 7.11T + 37T^{2} \) |
| 41 | \( 1 + 6.93T + 41T^{2} \) |
| 43 | \( 1 - 4.20T + 43T^{2} \) |
| 47 | \( 1 - 6.78T + 47T^{2} \) |
| 53 | \( 1 - 7.29T + 53T^{2} \) |
| 59 | \( 1 - 6.54T + 59T^{2} \) |
| 61 | \( 1 + 7.08T + 61T^{2} \) |
| 67 | \( 1 - 1.32T + 67T^{2} \) |
| 71 | \( 1 - 12.3T + 71T^{2} \) |
| 73 | \( 1 + 4.80T + 73T^{2} \) |
| 79 | \( 1 - 14.4T + 79T^{2} \) |
| 83 | \( 1 - 8.13T + 83T^{2} \) |
| 89 | \( 1 - 0.456T + 89T^{2} \) |
| 97 | \( 1 + 6.57T + 97T^{2} \) |
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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
−7.80554219157175761294747837033, −7.20063316414962832440556643136, −6.49950710118078356964698199138, −5.68223933517560026715427252748, −5.18120010842615137350545119067, −4.37547337511045704742910908433, −3.63429093355590442312252660817, −3.30875210142114847828463281392, −2.19942696842313804537932268815, −1.13748789484607988693499755703,
1.13748789484607988693499755703, 2.19942696842313804537932268815, 3.30875210142114847828463281392, 3.63429093355590442312252660817, 4.37547337511045704742910908433, 5.18120010842615137350545119067, 5.68223933517560026715427252748, 6.49950710118078356964698199138, 7.20063316414962832440556643136, 7.80554219157175761294747837033
