| L(s) = 1 | + 2-s − 0.289·3-s + 4-s − 3.81·5-s − 0.289·6-s − 2.88·7-s + 8-s − 2.91·9-s − 3.81·10-s + 3.27·11-s − 0.289·12-s + 6.15·13-s − 2.88·14-s + 1.10·15-s + 16-s − 0.420·17-s − 2.91·18-s + 3.90·19-s − 3.81·20-s + 0.833·21-s + 3.27·22-s + 6.10·23-s − 0.289·24-s + 9.56·25-s + 6.15·26-s + 1.71·27-s − 2.88·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.167·3-s + 0.5·4-s − 1.70·5-s − 0.118·6-s − 1.08·7-s + 0.353·8-s − 0.972·9-s − 1.20·10-s + 0.986·11-s − 0.0835·12-s + 1.70·13-s − 0.770·14-s + 0.285·15-s + 0.250·16-s − 0.101·17-s − 0.687·18-s + 0.896·19-s − 0.853·20-s + 0.181·21-s + 0.697·22-s + 1.27·23-s − 0.0590·24-s + 1.91·25-s + 1.20·26-s + 0.329·27-s − 0.544·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 2017 | \( 1 - T \) |
| good | 3 | \( 1 + 0.289T + 3T^{2} \) |
| 5 | \( 1 + 3.81T + 5T^{2} \) |
| 7 | \( 1 + 2.88T + 7T^{2} \) |
| 11 | \( 1 - 3.27T + 11T^{2} \) |
| 13 | \( 1 - 6.15T + 13T^{2} \) |
| 17 | \( 1 + 0.420T + 17T^{2} \) |
| 19 | \( 1 - 3.90T + 19T^{2} \) |
| 23 | \( 1 - 6.10T + 23T^{2} \) |
| 29 | \( 1 + 6.58T + 29T^{2} \) |
| 31 | \( 1 + 4.00T + 31T^{2} \) |
| 37 | \( 1 + 0.576T + 37T^{2} \) |
| 41 | \( 1 + 6.71T + 41T^{2} \) |
| 43 | \( 1 + 6.84T + 43T^{2} \) |
| 47 | \( 1 + 12.4T + 47T^{2} \) |
| 53 | \( 1 - 8.92T + 53T^{2} \) |
| 59 | \( 1 - 0.149T + 59T^{2} \) |
| 61 | \( 1 - 1.77T + 61T^{2} \) |
| 67 | \( 1 + 12.5T + 67T^{2} \) |
| 71 | \( 1 - 9.66T + 71T^{2} \) |
| 73 | \( 1 + 9.49T + 73T^{2} \) |
| 79 | \( 1 + 13.8T + 79T^{2} \) |
| 83 | \( 1 - 5.56T + 83T^{2} \) |
| 89 | \( 1 - 2.92T + 89T^{2} \) |
| 97 | \( 1 + 3.73T + 97T^{2} \) |
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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
−8.063145305152702934660343416352, −7.08811148572204434546238241624, −6.66043801269977538219541649397, −5.86585536246148795242522835489, −5.04989825485915069777523944752, −3.97883735375108582012451147293, −3.40014968019955193470089374832, −3.16880491098036887347128509413, −1.29724846033799191123354638289, 0,
1.29724846033799191123354638289, 3.16880491098036887347128509413, 3.40014968019955193470089374832, 3.97883735375108582012451147293, 5.04989825485915069777523944752, 5.86585536246148795242522835489, 6.66043801269977538219541649397, 7.08811148572204434546238241624, 8.063145305152702934660343416352
