| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.260298783\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.260298783\) |
| \(L(\frac{3}{2})\) |
|
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
−8.543771500015794953443947376918, −7.86586290019927462790060461986, −7.07489395834286858302410847165, −6.24981769626355019918652582624, −5.63402523752413484299138712560, −4.50553481623101326589650674566, −3.54788086351682203618161316426, −2.66264796377232142432182985042, −2.45549752603213072803367183950, −1.15830246043253114356725802787,
1.15830246043253114356725802787, 2.45549752603213072803367183950, 2.66264796377232142432182985042, 3.54788086351682203618161316426, 4.50553481623101326589650674566, 5.63402523752413484299138712560, 6.24981769626355019918652582624, 7.07489395834286858302410847165, 7.86586290019927462790060461986, 8.543771500015794953443947376918
