| L(s) = 1 | + 2.96·3-s + 0.884·5-s − 3.59·7-s + 5.79·9-s − 5.74·11-s − 0.730·13-s + 2.62·15-s + 1.55·17-s − 0.000218·19-s − 10.6·21-s + 5.06·23-s − 4.21·25-s + 8.28·27-s − 2.90·29-s + 2.03·31-s − 17.0·33-s − 3.18·35-s − 9.59·37-s − 2.16·39-s − 5.53·41-s − 5.49·43-s + 5.12·45-s − 4.78·47-s + 5.93·49-s + 4.61·51-s + 6.57·53-s − 5.08·55-s + ⋯ |
| L(s) = 1 | + 1.71·3-s + 0.395·5-s − 1.35·7-s + 1.93·9-s − 1.73·11-s − 0.202·13-s + 0.677·15-s + 0.377·17-s − 5.00e − 5·19-s − 2.32·21-s + 1.05·23-s − 0.843·25-s + 1.59·27-s − 0.539·29-s + 0.364·31-s − 2.96·33-s − 0.537·35-s − 1.57·37-s − 0.346·39-s − 0.864·41-s − 0.838·43-s + 0.764·45-s − 0.698·47-s + 0.847·49-s + 0.646·51-s + 0.903·53-s − 0.685·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 751 | \( 1 + T \) |
| good | 3 | \( 1 - 2.96T + 3T^{2} \) |
| 5 | \( 1 - 0.884T + 5T^{2} \) |
| 7 | \( 1 + 3.59T + 7T^{2} \) |
| 11 | \( 1 + 5.74T + 11T^{2} \) |
| 13 | \( 1 + 0.730T + 13T^{2} \) |
| 17 | \( 1 - 1.55T + 17T^{2} \) |
| 19 | \( 1 + 0.000218T + 19T^{2} \) |
| 23 | \( 1 - 5.06T + 23T^{2} \) |
| 29 | \( 1 + 2.90T + 29T^{2} \) |
| 31 | \( 1 - 2.03T + 31T^{2} \) |
| 37 | \( 1 + 9.59T + 37T^{2} \) |
| 41 | \( 1 + 5.53T + 41T^{2} \) |
| 43 | \( 1 + 5.49T + 43T^{2} \) |
| 47 | \( 1 + 4.78T + 47T^{2} \) |
| 53 | \( 1 - 6.57T + 53T^{2} \) |
| 59 | \( 1 + 10.1T + 59T^{2} \) |
| 61 | \( 1 + 0.100T + 61T^{2} \) |
| 67 | \( 1 + 13.6T + 67T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 73 | \( 1 - 12.8T + 73T^{2} \) |
| 79 | \( 1 - 12.1T + 79T^{2} \) |
| 83 | \( 1 - 4.78T + 83T^{2} \) |
| 89 | \( 1 + 14.9T + 89T^{2} \) |
| 97 | \( 1 - 2.87T + 97T^{2} \) |
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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
−7.76157650739730330072017754733, −7.24236047630112307396685816818, −6.49385908064027077439016072582, −5.52487748259192721093162848215, −4.79539760951132295818342475749, −3.59048136691348799727660778607, −3.15944635294387467944337490368, −2.55684185148236638351947305385, −1.71643318549574525985467568235, 0,
1.71643318549574525985467568235, 2.55684185148236638351947305385, 3.15944635294387467944337490368, 3.59048136691348799727660778607, 4.79539760951132295818342475749, 5.52487748259192721093162848215, 6.49385908064027077439016072582, 7.24236047630112307396685816818, 7.76157650739730330072017754733
