| L(s) = 1 | + 1.71·2-s − 1.49·3-s + 0.937·4-s − 2.05·5-s − 2.56·6-s + 4.47·7-s − 1.82·8-s − 0.763·9-s − 3.52·10-s − 1.03·11-s − 1.40·12-s + 7.66·14-s + 3.07·15-s − 4.99·16-s + 6.17·17-s − 1.30·18-s + 4.34·19-s − 1.92·20-s − 6.69·21-s − 1.78·22-s − 8.83·23-s + 2.72·24-s − 0.770·25-s + 5.62·27-s + 4.19·28-s + 2.44·29-s + 5.27·30-s + ⋯ |
| L(s) = 1 | + 1.21·2-s − 0.863·3-s + 0.468·4-s − 0.919·5-s − 1.04·6-s + 1.69·7-s − 0.643·8-s − 0.254·9-s − 1.11·10-s − 0.313·11-s − 0.404·12-s + 2.04·14-s + 0.794·15-s − 1.24·16-s + 1.49·17-s − 0.308·18-s + 0.997·19-s − 0.431·20-s − 1.45·21-s − 0.379·22-s − 1.84·23-s + 0.555·24-s − 0.154·25-s + 1.08·27-s + 0.792·28-s + 0.454·29-s + 0.962·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.084755902\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.084755902\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 - 1.71T + 2T^{2} \) |
| 3 | \( 1 + 1.49T + 3T^{2} \) |
| 5 | \( 1 + 2.05T + 5T^{2} \) |
| 7 | \( 1 - 4.47T + 7T^{2} \) |
| 11 | \( 1 + 1.03T + 11T^{2} \) |
| 17 | \( 1 - 6.17T + 17T^{2} \) |
| 19 | \( 1 - 4.34T + 19T^{2} \) |
| 23 | \( 1 + 8.83T + 23T^{2} \) |
| 29 | \( 1 - 2.44T + 29T^{2} \) |
| 37 | \( 1 + 8.47T + 37T^{2} \) |
| 41 | \( 1 - 2.45T + 41T^{2} \) |
| 43 | \( 1 + 5.21T + 43T^{2} \) |
| 47 | \( 1 - 1.43T + 47T^{2} \) |
| 53 | \( 1 + 4.22T + 53T^{2} \) |
| 59 | \( 1 - 7.95T + 59T^{2} \) |
| 61 | \( 1 - 4.34T + 61T^{2} \) |
| 67 | \( 1 - 1.88T + 67T^{2} \) |
| 71 | \( 1 - 11.7T + 71T^{2} \) |
| 73 | \( 1 - 4.11T + 73T^{2} \) |
| 79 | \( 1 - 1.27T + 79T^{2} \) |
| 83 | \( 1 - 0.800T + 83T^{2} \) |
| 89 | \( 1 - 14.3T + 89T^{2} \) |
| 97 | \( 1 + 14.5T + 97T^{2} \) |
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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
−8.069557242725551958505142871998, −7.52887512710173516737661738187, −6.51073609371298020989385337503, −5.54146980762000962318413039704, −5.35368167367605407202091642907, −4.67281659141607205981004080291, −3.89759677534439748476294375756, −3.22617854685895324028514128448, −2.00040847631067350995628452411, −0.67827943671508093794683424090,
0.67827943671508093794683424090, 2.00040847631067350995628452411, 3.22617854685895324028514128448, 3.89759677534439748476294375756, 4.67281659141607205981004080291, 5.35368167367605407202091642907, 5.54146980762000962318413039704, 6.51073609371298020989385337503, 7.52887512710173516737661738187, 8.069557242725551958505142871998
