| L(s) = 1 | + 1.42·2-s − 3-s + 0.0438·4-s + 5-s − 1.42·6-s − 4.47·7-s − 2.79·8-s + 9-s + 1.42·10-s − 2.86·11-s − 0.0438·12-s + 13-s − 6.39·14-s − 15-s − 4.08·16-s − 6.35·17-s + 1.42·18-s + 2.17·19-s + 0.0438·20-s + 4.47·21-s − 4.10·22-s + 1.22·23-s + 2.79·24-s + 25-s + 1.42·26-s − 27-s − 0.196·28-s + ⋯ |
| L(s) = 1 | + 1.01·2-s − 0.577·3-s + 0.0219·4-s + 0.447·5-s − 0.583·6-s − 1.69·7-s − 0.988·8-s + 0.333·9-s + 0.452·10-s − 0.865·11-s − 0.0126·12-s + 0.277·13-s − 1.70·14-s − 0.258·15-s − 1.02·16-s − 1.54·17-s + 0.336·18-s + 0.497·19-s + 0.00980·20-s + 0.976·21-s − 0.874·22-s + 0.255·23-s + 0.570·24-s + 0.200·25-s + 0.280·26-s − 0.192·27-s − 0.0370·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9613448398\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9613448398\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 - 1.42T + 2T^{2} \) |
| 7 | \( 1 + 4.47T + 7T^{2} \) |
| 11 | \( 1 + 2.86T + 11T^{2} \) |
| 17 | \( 1 + 6.35T + 17T^{2} \) |
| 19 | \( 1 - 2.17T + 19T^{2} \) |
| 23 | \( 1 - 1.22T + 23T^{2} \) |
| 29 | \( 1 + 3.42T + 29T^{2} \) |
| 37 | \( 1 + 1.26T + 37T^{2} \) |
| 41 | \( 1 + 8.39T + 41T^{2} \) |
| 43 | \( 1 - 6.39T + 43T^{2} \) |
| 47 | \( 1 + 4.53T + 47T^{2} \) |
| 53 | \( 1 + 9.61T + 53T^{2} \) |
| 59 | \( 1 + 6.33T + 59T^{2} \) |
| 61 | \( 1 - 1.51T + 61T^{2} \) |
| 67 | \( 1 - 13.7T + 67T^{2} \) |
| 71 | \( 1 - 7.05T + 71T^{2} \) |
| 73 | \( 1 - 11.0T + 73T^{2} \) |
| 79 | \( 1 + 9.26T + 79T^{2} \) |
| 83 | \( 1 - 15.6T + 83T^{2} \) |
| 89 | \( 1 + 9.24T + 89T^{2} \) |
| 97 | \( 1 - 17.8T + 97T^{2} \) |
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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
−8.011440393611777195828548887105, −6.77608735813697769437701959111, −6.61071586089673197438895240321, −5.83441370714140250171045548835, −5.25464853707703850534496750011, −4.55302870696664641400774387642, −3.64405360326184112830590678667, −3.04302009615087644694972550412, −2.16712098777596518745520643820, −0.42466991679852006864661321678,
0.42466991679852006864661321678, 2.16712098777596518745520643820, 3.04302009615087644694972550412, 3.64405360326184112830590678667, 4.55302870696664641400774387642, 5.25464853707703850534496750011, 5.83441370714140250171045548835, 6.61071586089673197438895240321, 6.77608735813697769437701959111, 8.011440393611777195828548887105
