L(s) = 1 | − 0.445·2-s − 1.80·4-s + 5-s − 0.554·7-s + 1.69·8-s − 0.445·10-s − 11-s + 0.246·14-s + 2.85·16-s + 3.10·17-s + 6.40·19-s − 1.80·20-s + 0.445·22-s − 3.24·23-s + 25-s + 28-s − 4.51·29-s − 7.09·31-s − 4.65·32-s − 1.38·34-s − 0.554·35-s − 3.58·37-s − 2.85·38-s + 1.69·40-s − 7.44·41-s − 5.44·43-s + 1.80·44-s + ⋯ |
L(s) = 1 | − 0.314·2-s − 0.900·4-s + 0.447·5-s − 0.209·7-s + 0.598·8-s − 0.140·10-s − 0.301·11-s + 0.0660·14-s + 0.712·16-s + 0.754·17-s + 1.46·19-s − 0.402·20-s + 0.0948·22-s − 0.677·23-s + 0.200·25-s + 0.188·28-s − 0.838·29-s − 1.27·31-s − 0.822·32-s − 0.237·34-s − 0.0938·35-s − 0.588·37-s − 0.462·38-s + 0.267·40-s − 1.16·41-s − 0.830·43-s + 0.271·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 0.445T + 2T^{2} \) |
| 7 | \( 1 + 0.554T + 7T^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 17 | \( 1 - 3.10T + 17T^{2} \) |
| 19 | \( 1 - 6.40T + 19T^{2} \) |
| 23 | \( 1 + 3.24T + 23T^{2} \) |
| 29 | \( 1 + 4.51T + 29T^{2} \) |
| 31 | \( 1 + 7.09T + 31T^{2} \) |
| 37 | \( 1 + 3.58T + 37T^{2} \) |
| 41 | \( 1 + 7.44T + 41T^{2} \) |
| 43 | \( 1 + 5.44T + 43T^{2} \) |
| 47 | \( 1 + 1.76T + 47T^{2} \) |
| 53 | \( 1 - 7.92T + 53T^{2} \) |
| 59 | \( 1 - 4.71T + 59T^{2} \) |
| 61 | \( 1 - 7.72T + 61T^{2} \) |
| 67 | \( 1 - 2.14T + 67T^{2} \) |
| 71 | \( 1 - 7.07T + 71T^{2} \) |
| 73 | \( 1 + 12.0T + 73T^{2} \) |
| 79 | \( 1 - 1.48T + 79T^{2} \) |
| 83 | \( 1 - 16.1T + 83T^{2} \) |
| 89 | \( 1 + 9.39T + 89T^{2} \) |
| 97 | \( 1 - 15.2T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.58616461123607427626895422642, −7.03481789169272162563647934184, −5.97706547212588188090392545911, −5.30872215802324084846326312382, −4.96533745892528000941714853364, −3.69920616155098527062632714554, −3.38760400805042818150113847936, −2.07270122922327625707397822267, −1.17297888499376732639342492099, 0,
1.17297888499376732639342492099, 2.07270122922327625707397822267, 3.38760400805042818150113847936, 3.69920616155098527062632714554, 4.96533745892528000941714853364, 5.30872215802324084846326312382, 5.97706547212588188090392545911, 7.03481789169272162563647934184, 7.58616461123607427626895422642