| L(s) = 1 | − 2.48·2-s + 4.16·4-s − 4.02·5-s + 2.71·7-s − 5.37·8-s + 9.99·10-s − 3.51·11-s − 6.31·13-s − 6.74·14-s + 5.02·16-s − 0.705·17-s + 19-s − 16.7·20-s + 8.71·22-s + 0.522·23-s + 11.1·25-s + 15.6·26-s + 11.3·28-s − 0.738·29-s + 1.19·31-s − 1.71·32-s + 1.75·34-s − 10.9·35-s − 5.41·37-s − 2.48·38-s + 21.6·40-s + 3.75·41-s + ⋯ |
| L(s) = 1 | − 1.75·2-s + 2.08·4-s − 1.79·5-s + 1.02·7-s − 1.90·8-s + 3.15·10-s − 1.05·11-s − 1.75·13-s − 1.80·14-s + 1.25·16-s − 0.171·17-s + 0.229·19-s − 3.74·20-s + 1.85·22-s + 0.108·23-s + 2.23·25-s + 3.07·26-s + 2.13·28-s − 0.137·29-s + 0.215·31-s − 0.303·32-s + 0.300·34-s − 1.84·35-s − 0.890·37-s − 0.402·38-s + 3.42·40-s + 0.585·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 - T \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 + 2.48T + 2T^{2} \) |
| 5 | \( 1 + 4.02T + 5T^{2} \) |
| 7 | \( 1 - 2.71T + 7T^{2} \) |
| 11 | \( 1 + 3.51T + 11T^{2} \) |
| 13 | \( 1 + 6.31T + 13T^{2} \) |
| 17 | \( 1 + 0.705T + 17T^{2} \) |
| 23 | \( 1 - 0.522T + 23T^{2} \) |
| 29 | \( 1 + 0.738T + 29T^{2} \) |
| 31 | \( 1 - 1.19T + 31T^{2} \) |
| 37 | \( 1 + 5.41T + 37T^{2} \) |
| 41 | \( 1 - 3.75T + 41T^{2} \) |
| 43 | \( 1 + 3.71T + 43T^{2} \) |
| 53 | \( 1 - 2.32T + 53T^{2} \) |
| 59 | \( 1 - 2.47T + 59T^{2} \) |
| 61 | \( 1 - 5.98T + 61T^{2} \) |
| 67 | \( 1 - 2.42T + 67T^{2} \) |
| 71 | \( 1 - 13.2T + 71T^{2} \) |
| 73 | \( 1 - 12.4T + 73T^{2} \) |
| 79 | \( 1 - 5.81T + 79T^{2} \) |
| 83 | \( 1 - 7.81T + 83T^{2} \) |
| 89 | \( 1 - 5.89T + 89T^{2} \) |
| 97 | \( 1 - 1.48T + 97T^{2} \) |
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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
−7.79197310420951146075376099565, −7.22931190024114138944732510913, −6.70172373610526795677966209060, −5.17837015642353085486617567898, −4.83606055390426360476403059587, −3.77881353551748695371696566929, −2.73436132819165438065420139649, −2.08062652538415552856779655353, −0.790383612224058172779977744056, 0,
0.790383612224058172779977744056, 2.08062652538415552856779655353, 2.73436132819165438065420139649, 3.77881353551748695371696566929, 4.83606055390426360476403059587, 5.17837015642353085486617567898, 6.70172373610526795677966209060, 7.22931190024114138944732510913, 7.79197310420951146075376099565
