L(s) = 1 | + 2.24·2-s − 2.24·3-s + 3.04·4-s − 2.35·5-s − 5.04·6-s + 2.69·7-s + 2.35·8-s + 2.04·9-s − 5.29·10-s + 0.109·11-s − 6.85·12-s + 6.04·14-s + 5.29·15-s − 0.801·16-s − 0.554·17-s + 4.60·18-s + 19-s − 7.18·20-s − 6.04·21-s + 0.246·22-s − 0.801·23-s − 5.29·24-s + 0.554·25-s + 2.13·27-s + 8.20·28-s + 3.35·29-s + 11.8·30-s + ⋯ |
L(s) = 1 | + 1.58·2-s − 1.29·3-s + 1.52·4-s − 1.05·5-s − 2.06·6-s + 1.01·7-s + 0.833·8-s + 0.682·9-s − 1.67·10-s + 0.0331·11-s − 1.97·12-s + 1.61·14-s + 1.36·15-s − 0.200·16-s − 0.134·17-s + 1.08·18-s + 0.229·19-s − 1.60·20-s − 1.31·21-s + 0.0526·22-s − 0.167·23-s − 1.08·24-s + 0.110·25-s + 0.411·27-s + 1.55·28-s + 0.623·29-s + 2.17·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 19 | \( 1 - T \) |
| 211 | \( 1 - T \) |
good | 2 | \( 1 - 2.24T + 2T^{2} \) |
| 3 | \( 1 + 2.24T + 3T^{2} \) |
| 5 | \( 1 + 2.35T + 5T^{2} \) |
| 7 | \( 1 - 2.69T + 7T^{2} \) |
| 11 | \( 1 - 0.109T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 0.554T + 17T^{2} \) |
| 23 | \( 1 + 0.801T + 23T^{2} \) |
| 29 | \( 1 - 3.35T + 29T^{2} \) |
| 31 | \( 1 - 4.26T + 31T^{2} \) |
| 37 | \( 1 + 7.50T + 37T^{2} \) |
| 41 | \( 1 - 0.268T + 41T^{2} \) |
| 43 | \( 1 + 3.49T + 43T^{2} \) |
| 47 | \( 1 + 11.8T + 47T^{2} \) |
| 53 | \( 1 + 7.64T + 53T^{2} \) |
| 59 | \( 1 - 4.18T + 59T^{2} \) |
| 61 | \( 1 - 4.49T + 61T^{2} \) |
| 67 | \( 1 - 11.0T + 67T^{2} \) |
| 71 | \( 1 + 5.52T + 71T^{2} \) |
| 73 | \( 1 - 8.74T + 73T^{2} \) |
| 79 | \( 1 + 9.64T + 79T^{2} \) |
| 83 | \( 1 - 5.34T + 83T^{2} \) |
| 89 | \( 1 + 18.5T + 89T^{2} \) |
| 97 | \( 1 - 2.29T + 97T^{2} \) |
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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
−7.959542409236452173542121465243, −6.85249638898217771498435458475, −6.54178302240384191757751189911, −5.48294246294033628407558444557, −5.08002618641658441607554371738, −4.46103528426471747673282487134, −3.79572104337068078377712193302, −2.82597799379438794977861726925, −1.49501662985074360649748489447, 0,
1.49501662985074360649748489447, 2.82597799379438794977861726925, 3.79572104337068078377712193302, 4.46103528426471747673282487134, 5.08002618641658441607554371738, 5.48294246294033628407558444557, 6.54178302240384191757751189911, 6.85249638898217771498435458475, 7.959542409236452173542121465243