| L(s) = 1 | + 2.30·5-s + 7-s + 2.47·11-s − 6.18·13-s − 1.60·17-s + 7.35·19-s + 23-s + 0.302·25-s − 9.96·29-s + 3.08·31-s + 2.30·35-s + 7.88·37-s + 0.119·41-s + 2.50·43-s + 49-s + 11.7·53-s + 5.70·55-s + 9.98·59-s + 1.82·61-s − 14.2·65-s − 5.78·67-s + 10.7·71-s + 13.3·73-s + 2.47·77-s − 4.92·79-s − 4.15·83-s − 3.69·85-s + ⋯ |
| L(s) = 1 | + 1.02·5-s + 0.377·7-s + 0.747·11-s − 1.71·13-s − 0.389·17-s + 1.68·19-s + 0.208·23-s + 0.0605·25-s − 1.85·29-s + 0.553·31-s + 0.389·35-s + 1.29·37-s + 0.0187·41-s + 0.382·43-s + 0.142·49-s + 1.60·53-s + 0.769·55-s + 1.29·59-s + 0.233·61-s − 1.76·65-s − 0.706·67-s + 1.27·71-s + 1.56·73-s + 0.282·77-s − 0.554·79-s − 0.456·83-s − 0.401·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5796 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5796 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.593156418\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.593156418\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 - 2.30T + 5T^{2} \) |
| 11 | \( 1 - 2.47T + 11T^{2} \) |
| 13 | \( 1 + 6.18T + 13T^{2} \) |
| 17 | \( 1 + 1.60T + 17T^{2} \) |
| 19 | \( 1 - 7.35T + 19T^{2} \) |
| 29 | \( 1 + 9.96T + 29T^{2} \) |
| 31 | \( 1 - 3.08T + 31T^{2} \) |
| 37 | \( 1 - 7.88T + 37T^{2} \) |
| 41 | \( 1 - 0.119T + 41T^{2} \) |
| 43 | \( 1 - 2.50T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 11.7T + 53T^{2} \) |
| 59 | \( 1 - 9.98T + 59T^{2} \) |
| 61 | \( 1 - 1.82T + 61T^{2} \) |
| 67 | \( 1 + 5.78T + 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 - 13.3T + 73T^{2} \) |
| 79 | \( 1 + 4.92T + 79T^{2} \) |
| 83 | \( 1 + 4.15T + 83T^{2} \) |
| 89 | \( 1 - 11.4T + 89T^{2} \) |
| 97 | \( 1 + 7.15T + 97T^{2} \) |
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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
−7.985857526358089948672371655672, −7.33681808990304168235227604021, −6.80431082126559183493790048710, −5.77602413020970294531152590853, −5.36648604379174905098119616009, −4.58665264629332099372511682407, −3.67059346057768992440626545929, −2.56363059771592394097952727116, −1.98269561939272075408751270960, −0.868437637948494082719718296652,
0.868437637948494082719718296652, 1.98269561939272075408751270960, 2.56363059771592394097952727116, 3.67059346057768992440626545929, 4.58665264629332099372511682407, 5.36648604379174905098119616009, 5.77602413020970294531152590853, 6.80431082126559183493790048710, 7.33681808990304168235227604021, 7.985857526358089948672371655672
