L(s) = 1 | − 3-s + 1.25·5-s + 4.55·7-s + 9-s + 5.23·11-s − 1.93·13-s − 1.25·15-s + 7.59·17-s + 4.42·19-s − 4.55·21-s − 6.61·23-s − 3.42·25-s − 27-s + 3.73·29-s + 1.27·31-s − 5.23·33-s + 5.72·35-s + 6.87·37-s + 1.93·39-s + 8.71·41-s − 5.25·43-s + 1.25·45-s + 6.55·47-s + 13.7·49-s − 7.59·51-s + 3.93·53-s + 6.57·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.562·5-s + 1.72·7-s + 0.333·9-s + 1.57·11-s − 0.536·13-s − 0.324·15-s + 1.84·17-s + 1.01·19-s − 0.994·21-s − 1.37·23-s − 0.684·25-s − 0.192·27-s + 0.693·29-s + 0.228·31-s − 0.911·33-s + 0.967·35-s + 1.12·37-s + 0.309·39-s + 1.36·41-s − 0.801·43-s + 0.187·45-s + 0.956·47-s + 1.96·49-s − 1.06·51-s + 0.541·53-s + 0.887·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.089625486\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.089625486\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 167 | \( 1 - T \) |
good | 5 | \( 1 - 1.25T + 5T^{2} \) |
| 7 | \( 1 - 4.55T + 7T^{2} \) |
| 11 | \( 1 - 5.23T + 11T^{2} \) |
| 13 | \( 1 + 1.93T + 13T^{2} \) |
| 17 | \( 1 - 7.59T + 17T^{2} \) |
| 19 | \( 1 - 4.42T + 19T^{2} \) |
| 23 | \( 1 + 6.61T + 23T^{2} \) |
| 29 | \( 1 - 3.73T + 29T^{2} \) |
| 31 | \( 1 - 1.27T + 31T^{2} \) |
| 37 | \( 1 - 6.87T + 37T^{2} \) |
| 41 | \( 1 - 8.71T + 41T^{2} \) |
| 43 | \( 1 + 5.25T + 43T^{2} \) |
| 47 | \( 1 - 6.55T + 47T^{2} \) |
| 53 | \( 1 - 3.93T + 53T^{2} \) |
| 59 | \( 1 + 12.3T + 59T^{2} \) |
| 61 | \( 1 + 0.282T + 61T^{2} \) |
| 67 | \( 1 - 8.25T + 67T^{2} \) |
| 71 | \( 1 - 1.66T + 71T^{2} \) |
| 73 | \( 1 + 3.89T + 73T^{2} \) |
| 79 | \( 1 + 6.21T + 79T^{2} \) |
| 83 | \( 1 + 14.1T + 83T^{2} \) |
| 89 | \( 1 + 6.93T + 89T^{2} \) |
| 97 | \( 1 + 3.11T + 97T^{2} \) |
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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
−7.66235933844664403446869451286, −7.37016514997880932859308690721, −6.16711922434182262978075686903, −5.81847990393578782625349332065, −5.09559686488064618034723459547, −4.38544475097262349176748740082, −3.73175290795900905548652163434, −2.47827136972535191418501082054, −1.46813786678845969533585755582, −1.07434519172689328599194454799,
1.07434519172689328599194454799, 1.46813786678845969533585755582, 2.47827136972535191418501082054, 3.73175290795900905548652163434, 4.38544475097262349176748740082, 5.09559686488064618034723459547, 5.81847990393578782625349332065, 6.16711922434182262978075686903, 7.37016514997880932859308690721, 7.66235933844664403446869451286