| L(s) = 1 | − 3-s + 5-s + 3.12·7-s + 9-s − 2·11-s − 3.12·13-s − 15-s + 7.12·17-s + 3.12·19-s − 3.12·21-s − 3.12·23-s + 25-s − 27-s + 8.24·29-s − 1.12·31-s + 2·33-s + 3.12·35-s + 3.12·37-s + 3.12·39-s − 2·41-s − 10.2·43-s + 45-s + 4.87·47-s + 2.75·49-s − 7.12·51-s + 10·53-s − 2·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.18·7-s + 0.333·9-s − 0.603·11-s − 0.866·13-s − 0.258·15-s + 1.72·17-s + 0.716·19-s − 0.681·21-s − 0.651·23-s + 0.200·25-s − 0.192·27-s + 1.53·29-s − 0.201·31-s + 0.348·33-s + 0.527·35-s + 0.513·37-s + 0.500·39-s − 0.312·41-s − 1.56·43-s + 0.149·45-s + 0.711·47-s + 0.393·49-s − 0.997·51-s + 1.37·53-s − 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.799119137\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.799119137\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| good | 7 | \( 1 - 3.12T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 3.12T + 13T^{2} \) |
| 17 | \( 1 - 7.12T + 17T^{2} \) |
| 19 | \( 1 - 3.12T + 19T^{2} \) |
| 23 | \( 1 + 3.12T + 23T^{2} \) |
| 29 | \( 1 - 8.24T + 29T^{2} \) |
| 31 | \( 1 + 1.12T + 31T^{2} \) |
| 37 | \( 1 - 3.12T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 10.2T + 43T^{2} \) |
| 47 | \( 1 - 4.87T + 47T^{2} \) |
| 53 | \( 1 - 10T + 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 10.2T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 12.2T + 73T^{2} \) |
| 79 | \( 1 - 13.1T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−9.356791594914383908210474172565, −8.143208932893733349944922815051, −7.78009946351911600510911295873, −6.86663253534215083547353363129, −5.78356093226106515295114625768, −5.19060275857337787418167720091, −4.61313099545740155348448603371, −3.25433193910272785317620247166, −2.09642406181533508831481710746, −0.979120241433238298053492313580,
0.979120241433238298053492313580, 2.09642406181533508831481710746, 3.25433193910272785317620247166, 4.61313099545740155348448603371, 5.19060275857337787418167720091, 5.78356093226106515295114625768, 6.86663253534215083547353363129, 7.78009946351911600510911295873, 8.143208932893733349944922815051, 9.356791594914383908210474172565
