| L(s) = 1 | − 0.976·2-s − 3-s − 1.04·4-s − 5-s + 0.976·6-s + 2.96·7-s + 2.97·8-s + 9-s + 0.976·10-s − 4.81·11-s + 1.04·12-s − 13-s − 2.89·14-s + 15-s − 0.813·16-s − 6.91·17-s − 0.976·18-s − 7.22·19-s + 1.04·20-s − 2.96·21-s + 4.70·22-s − 8.09·23-s − 2.97·24-s + 25-s + 0.976·26-s − 27-s − 3.09·28-s + ⋯ |
| L(s) = 1 | − 0.690·2-s − 0.577·3-s − 0.523·4-s − 0.447·5-s + 0.398·6-s + 1.11·7-s + 1.05·8-s + 0.333·9-s + 0.308·10-s − 1.45·11-s + 0.301·12-s − 0.277·13-s − 0.772·14-s + 0.258·15-s − 0.203·16-s − 1.67·17-s − 0.230·18-s − 1.65·19-s + 0.233·20-s − 0.646·21-s + 1.00·22-s − 1.68·23-s − 0.607·24-s + 0.200·25-s + 0.191·26-s − 0.192·27-s − 0.585·28-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
| \(L(1)\) |
\(\approx\) |
\(0.1565386235\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1565386235\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.976T + 2T^{2} \) |
| 7 | \( 1 - 2.96T + 7T^{2} \) |
| 11 | \( 1 + 4.81T + 11T^{2} \) |
| 17 | \( 1 + 6.91T + 17T^{2} \) |
| 19 | \( 1 + 7.22T + 19T^{2} \) |
| 23 | \( 1 + 8.09T + 23T^{2} \) |
| 29 | \( 1 + 7.96T + 29T^{2} \) |
| 37 | \( 1 - 7.72T + 37T^{2} \) |
| 41 | \( 1 + 1.33T + 41T^{2} \) |
| 43 | \( 1 + 0.349T + 43T^{2} \) |
| 47 | \( 1 - 2.48T + 47T^{2} \) |
| 53 | \( 1 + 4.30T + 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 + 6.25T + 61T^{2} \) |
| 67 | \( 1 - 4.18T + 67T^{2} \) |
| 71 | \( 1 + 13.4T + 71T^{2} \) |
| 73 | \( 1 - 9.74T + 73T^{2} \) |
| 79 | \( 1 + 15.9T + 79T^{2} \) |
| 83 | \( 1 - 9.09T + 83T^{2} \) |
| 89 | \( 1 + 5.47T + 89T^{2} \) |
| 97 | \( 1 + 1.05T + 97T^{2} \) |
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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
−8.083363156217839227384986704773, −7.68753215559229926318067455120, −6.82767444700821777437607225876, −5.85736275610848966037452347174, −5.11990340435520829490029868494, −4.34704823013003162613641276289, −4.14312578757449455561423335187, −2.38841862573963860346617749296, −1.76876574782501586437808203706, −0.23141376447649094822619378464,
0.23141376447649094822619378464, 1.76876574782501586437808203706, 2.38841862573963860346617749296, 4.14312578757449455561423335187, 4.34704823013003162613641276289, 5.11990340435520829490029868494, 5.85736275610848966037452347174, 6.82767444700821777437607225876, 7.68753215559229926318067455120, 8.083363156217839227384986704773
