| L(s) = 1 | + 2-s − 1.70·3-s + 4-s − 0.390·5-s − 1.70·6-s − 1.43·7-s + 8-s − 0.0932·9-s − 0.390·10-s − 1.03·11-s − 1.70·12-s − 3.10·13-s − 1.43·14-s + 0.665·15-s + 16-s + 0.487·17-s − 0.0932·18-s + 7.55·19-s − 0.390·20-s + 2.44·21-s − 1.03·22-s + 8.23·23-s − 1.70·24-s − 4.84·25-s − 3.10·26-s + 5.27·27-s − 1.43·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.984·3-s + 0.5·4-s − 0.174·5-s − 0.696·6-s − 0.542·7-s + 0.353·8-s − 0.0310·9-s − 0.123·10-s − 0.310·11-s − 0.492·12-s − 0.859·13-s − 0.383·14-s + 0.171·15-s + 0.250·16-s + 0.118·17-s − 0.0219·18-s + 1.73·19-s − 0.0873·20-s + 0.534·21-s − 0.219·22-s + 1.71·23-s − 0.348·24-s − 0.969·25-s − 0.607·26-s + 1.01·27-s − 0.271·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 2003 | \( 1 - T \) |
| good | 3 | \( 1 + 1.70T + 3T^{2} \) |
| 5 | \( 1 + 0.390T + 5T^{2} \) |
| 7 | \( 1 + 1.43T + 7T^{2} \) |
| 11 | \( 1 + 1.03T + 11T^{2} \) |
| 13 | \( 1 + 3.10T + 13T^{2} \) |
| 17 | \( 1 - 0.487T + 17T^{2} \) |
| 19 | \( 1 - 7.55T + 19T^{2} \) |
| 23 | \( 1 - 8.23T + 23T^{2} \) |
| 29 | \( 1 - 6.02T + 29T^{2} \) |
| 31 | \( 1 + 0.353T + 31T^{2} \) |
| 37 | \( 1 + 2.90T + 37T^{2} \) |
| 41 | \( 1 - 5.19T + 41T^{2} \) |
| 43 | \( 1 + 7.72T + 43T^{2} \) |
| 47 | \( 1 + 1.80T + 47T^{2} \) |
| 53 | \( 1 + 6.12T + 53T^{2} \) |
| 59 | \( 1 + 9.57T + 59T^{2} \) |
| 61 | \( 1 + 14.3T + 61T^{2} \) |
| 67 | \( 1 + 1.86T + 67T^{2} \) |
| 71 | \( 1 + 13.7T + 71T^{2} \) |
| 73 | \( 1 + 7.41T + 73T^{2} \) |
| 79 | \( 1 + 1.57T + 79T^{2} \) |
| 83 | \( 1 - 1.37T + 83T^{2} \) |
| 89 | \( 1 + 6.46T + 89T^{2} \) |
| 97 | \( 1 - 0.813T + 97T^{2} \) |
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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
−7.73206662136389021085671870856, −7.23567044767084479232921304852, −6.41889912464699039587811181198, −5.79454293224303903099253587392, −5.00941170053335686232949532351, −4.65732696527192053379191767026, −3.21801413117210633854910830337, −2.90389521974578044442320617634, −1.32102270043779538756075306207, 0,
1.32102270043779538756075306207, 2.90389521974578044442320617634, 3.21801413117210633854910830337, 4.65732696527192053379191767026, 5.00941170053335686232949532351, 5.79454293224303903099253587392, 6.41889912464699039587811181198, 7.23567044767084479232921304852, 7.73206662136389021085671870856
