| L(s) = 1 | + 2.36·3-s + 5-s + 2.16·7-s + 2.58·9-s + 1.75·11-s + 2.60·13-s + 2.36·15-s − 5.49·17-s + 5.90·19-s + 5.12·21-s − 7.29·23-s + 25-s − 0.978·27-s − 29-s − 0.169·31-s + 4.15·33-s + 2.16·35-s + 3.75·37-s + 6.16·39-s − 10.3·41-s + 4.55·43-s + 2.58·45-s + 1.17·47-s − 2.29·49-s − 12.9·51-s + 11.8·53-s + 1.75·55-s + ⋯ |
| L(s) = 1 | + 1.36·3-s + 0.447·5-s + 0.820·7-s + 0.862·9-s + 0.529·11-s + 0.723·13-s + 0.610·15-s − 1.33·17-s + 1.35·19-s + 1.11·21-s − 1.52·23-s + 0.200·25-s − 0.188·27-s − 0.185·29-s − 0.0305·31-s + 0.722·33-s + 0.366·35-s + 0.617·37-s + 0.986·39-s − 1.62·41-s + 0.694·43-s + 0.385·45-s + 0.171·47-s − 0.327·49-s − 1.81·51-s + 1.62·53-s + 0.236·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.098549375\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.098549375\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 - 2.36T + 3T^{2} \) |
| 7 | \( 1 - 2.16T + 7T^{2} \) |
| 11 | \( 1 - 1.75T + 11T^{2} \) |
| 13 | \( 1 - 2.60T + 13T^{2} \) |
| 17 | \( 1 + 5.49T + 17T^{2} \) |
| 19 | \( 1 - 5.90T + 19T^{2} \) |
| 23 | \( 1 + 7.29T + 23T^{2} \) |
| 31 | \( 1 + 0.169T + 31T^{2} \) |
| 37 | \( 1 - 3.75T + 37T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 - 4.55T + 43T^{2} \) |
| 47 | \( 1 - 1.17T + 47T^{2} \) |
| 53 | \( 1 - 11.8T + 53T^{2} \) |
| 59 | \( 1 + 4.11T + 59T^{2} \) |
| 61 | \( 1 - 1.00T + 61T^{2} \) |
| 67 | \( 1 + 8.48T + 67T^{2} \) |
| 71 | \( 1 - 6.10T + 71T^{2} \) |
| 73 | \( 1 + 1.72T + 73T^{2} \) |
| 79 | \( 1 + 2.02T + 79T^{2} \) |
| 83 | \( 1 - 1.86T + 83T^{2} \) |
| 89 | \( 1 - 16.5T + 89T^{2} \) |
| 97 | \( 1 - 0.464T + 97T^{2} \) |
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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
−9.526575145113876665310531888466, −8.930590367603212666255588462952, −8.266399900780545732894954577342, −7.57316114623749431934838096163, −6.54995551773627559645820167044, −5.53786481669365680781751263131, −4.36110763073305416691484035441, −3.54343183443131347105662667459, −2.38600569989539124294696129828, −1.53198318254994865028898667172,
1.53198318254994865028898667172, 2.38600569989539124294696129828, 3.54343183443131347105662667459, 4.36110763073305416691484035441, 5.53786481669365680781751263131, 6.54995551773627559645820167044, 7.57316114623749431934838096163, 8.266399900780545732894954577342, 8.930590367603212666255588462952, 9.526575145113876665310531888466
