L(s) = 1 | − 2.93·3-s + 3.00·5-s + 2.93·7-s + 5.61·9-s + 3.17·11-s + 1.63·13-s − 8.83·15-s + 0.761·17-s − 1.51·19-s − 8.62·21-s + 0.900·23-s + 4.05·25-s − 7.67·27-s + 4.23·29-s − 8.00·31-s − 9.30·33-s + 8.84·35-s + 0.300·37-s − 4.79·39-s + 12.4·41-s − 3.99·43-s + 16.8·45-s + 2.96·47-s + 1.63·49-s − 2.23·51-s + 10.0·53-s + 9.54·55-s + ⋯ |
L(s) = 1 | − 1.69·3-s + 1.34·5-s + 1.11·7-s + 1.87·9-s + 0.956·11-s + 0.452·13-s − 2.28·15-s + 0.184·17-s − 0.347·19-s − 1.88·21-s + 0.187·23-s + 0.810·25-s − 1.47·27-s + 0.785·29-s − 1.43·31-s − 1.62·33-s + 1.49·35-s + 0.0493·37-s − 0.767·39-s + 1.94·41-s − 0.609·43-s + 2.51·45-s + 0.432·47-s + 0.234·49-s − 0.312·51-s + 1.38·53-s + 1.28·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.088116768\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.088116768\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 503 | \( 1 - T \) |
good | 3 | \( 1 + 2.93T + 3T^{2} \) |
| 5 | \( 1 - 3.00T + 5T^{2} \) |
| 7 | \( 1 - 2.93T + 7T^{2} \) |
| 11 | \( 1 - 3.17T + 11T^{2} \) |
| 13 | \( 1 - 1.63T + 13T^{2} \) |
| 17 | \( 1 - 0.761T + 17T^{2} \) |
| 19 | \( 1 + 1.51T + 19T^{2} \) |
| 23 | \( 1 - 0.900T + 23T^{2} \) |
| 29 | \( 1 - 4.23T + 29T^{2} \) |
| 31 | \( 1 + 8.00T + 31T^{2} \) |
| 37 | \( 1 - 0.300T + 37T^{2} \) |
| 41 | \( 1 - 12.4T + 41T^{2} \) |
| 43 | \( 1 + 3.99T + 43T^{2} \) |
| 47 | \( 1 - 2.96T + 47T^{2} \) |
| 53 | \( 1 - 10.0T + 53T^{2} \) |
| 59 | \( 1 + 1.94T + 59T^{2} \) |
| 61 | \( 1 - 3.16T + 61T^{2} \) |
| 67 | \( 1 - 3.68T + 67T^{2} \) |
| 71 | \( 1 - 5.44T + 71T^{2} \) |
| 73 | \( 1 - 3.11T + 73T^{2} \) |
| 79 | \( 1 + 0.450T + 79T^{2} \) |
| 83 | \( 1 - 3.31T + 83T^{2} \) |
| 89 | \( 1 + 0.156T + 89T^{2} \) |
| 97 | \( 1 + 18.2T + 97T^{2} \) |
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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
−7.62128561971014210659925100802, −6.85040349131820249954148705856, −6.28142433463428796053698100555, −5.72767366983888757761266757454, −5.25715739597075012663697192106, −4.55493540559993283424210857126, −3.81659347656807928524774527036, −2.29634703599479790648267845586, −1.51103900503090724694221298749, −0.876456165048534485666231230512,
0.876456165048534485666231230512, 1.51103900503090724694221298749, 2.29634703599479790648267845586, 3.81659347656807928524774527036, 4.55493540559993283424210857126, 5.25715739597075012663697192106, 5.72767366983888757761266757454, 6.28142433463428796053698100555, 6.85040349131820249954148705856, 7.62128561971014210659925100802