L(s) = 1 | + 1.67·2-s + 0.796·4-s − 5-s + 3.27·7-s − 2.01·8-s − 1.67·10-s − 3.73·11-s + 1.23·13-s + 5.48·14-s − 4.95·16-s − 5.80·17-s + 6.13·19-s − 0.796·20-s − 6.25·22-s − 0.855·23-s + 25-s + 2.06·26-s + 2.61·28-s − 7.11·29-s − 1.95·31-s − 4.26·32-s − 9.71·34-s − 3.27·35-s − 7.33·37-s + 10.2·38-s + 2.01·40-s + 3.81·41-s + ⋯ |
L(s) = 1 | + 1.18·2-s + 0.398·4-s − 0.447·5-s + 1.23·7-s − 0.711·8-s − 0.528·10-s − 1.12·11-s + 0.342·13-s + 1.46·14-s − 1.23·16-s − 1.40·17-s + 1.40·19-s − 0.178·20-s − 1.33·22-s − 0.178·23-s + 0.200·25-s + 0.404·26-s + 0.493·28-s − 1.32·29-s − 0.351·31-s − 0.753·32-s − 1.66·34-s − 0.554·35-s − 1.20·37-s + 1.66·38-s + 0.318·40-s + 0.595·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 89 | \( 1 - T \) |
good | 2 | \( 1 - 1.67T + 2T^{2} \) |
| 7 | \( 1 - 3.27T + 7T^{2} \) |
| 11 | \( 1 + 3.73T + 11T^{2} \) |
| 13 | \( 1 - 1.23T + 13T^{2} \) |
| 17 | \( 1 + 5.80T + 17T^{2} \) |
| 19 | \( 1 - 6.13T + 19T^{2} \) |
| 23 | \( 1 + 0.855T + 23T^{2} \) |
| 29 | \( 1 + 7.11T + 29T^{2} \) |
| 31 | \( 1 + 1.95T + 31T^{2} \) |
| 37 | \( 1 + 7.33T + 37T^{2} \) |
| 41 | \( 1 - 3.81T + 41T^{2} \) |
| 43 | \( 1 + 7.67T + 43T^{2} \) |
| 47 | \( 1 - 9.67T + 47T^{2} \) |
| 53 | \( 1 + 0.0395T + 53T^{2} \) |
| 59 | \( 1 + 12.1T + 59T^{2} \) |
| 61 | \( 1 + 8.20T + 61T^{2} \) |
| 67 | \( 1 + 10.0T + 67T^{2} \) |
| 71 | \( 1 + 10.8T + 71T^{2} \) |
| 73 | \( 1 - 4.63T + 73T^{2} \) |
| 79 | \( 1 - 8.19T + 79T^{2} \) |
| 83 | \( 1 - 15.0T + 83T^{2} \) |
| 97 | \( 1 + 1.59T + 97T^{2} \) |
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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
−7.84681137586391205121251943139, −7.45400902203652970413972176984, −6.42146555852994129011095251823, −5.47505928507916244921656320456, −5.06740078208897870480163764051, −4.38526827905232247655756212674, −3.60078119781427416075983572314, −2.71913051344696855004984122539, −1.70841320981674659886003206055, 0,
1.70841320981674659886003206055, 2.71913051344696855004984122539, 3.60078119781427416075983572314, 4.38526827905232247655756212674, 5.06740078208897870480163764051, 5.47505928507916244921656320456, 6.42146555852994129011095251823, 7.45400902203652970413972176984, 7.84681137586391205121251943139