L(s) = 1 | − 2-s + 4-s − 7-s − 8-s − 5.29·11-s − 0.645·13-s + 14-s + 16-s + 17-s + 4.29·19-s + 5.29·22-s + 3.64·23-s + 0.645·26-s − 28-s − 5.35·29-s + 3.29·31-s − 32-s − 34-s + 4.35·37-s − 4.29·38-s − 3.29·41-s − 5.64·43-s − 5.29·44-s − 3.64·46-s + 4.64·47-s + 49-s − 0.645·52-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.377·7-s − 0.353·8-s − 1.59·11-s − 0.179·13-s + 0.267·14-s + 0.250·16-s + 0.242·17-s + 0.984·19-s + 1.12·22-s + 0.760·23-s + 0.126·26-s − 0.188·28-s − 0.994·29-s + 0.591·31-s − 0.176·32-s − 0.171·34-s + 0.715·37-s − 0.696·38-s − 0.514·41-s − 0.860·43-s − 0.797·44-s − 0.537·46-s + 0.677·47-s + 0.142·49-s − 0.0895·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 + 5.29T + 11T^{2} \) |
| 13 | \( 1 + 0.645T + 13T^{2} \) |
| 17 | \( 1 - T + 17T^{2} \) |
| 19 | \( 1 - 4.29T + 19T^{2} \) |
| 23 | \( 1 - 3.64T + 23T^{2} \) |
| 29 | \( 1 + 5.35T + 29T^{2} \) |
| 31 | \( 1 - 3.29T + 31T^{2} \) |
| 37 | \( 1 - 4.35T + 37T^{2} \) |
| 41 | \( 1 + 3.29T + 41T^{2} \) |
| 43 | \( 1 + 5.64T + 43T^{2} \) |
| 47 | \( 1 - 4.64T + 47T^{2} \) |
| 53 | \( 1 - 2.64T + 53T^{2} \) |
| 59 | \( 1 + 2.70T + 59T^{2} \) |
| 61 | \( 1 + 2.64T + 61T^{2} \) |
| 67 | \( 1 - 10.9T + 67T^{2} \) |
| 71 | \( 1 + 0.708T + 71T^{2} \) |
| 73 | \( 1 + 5.29T + 73T^{2} \) |
| 79 | \( 1 + 0.0627T + 79T^{2} \) |
| 83 | \( 1 - 1.64T + 83T^{2} \) |
| 89 | \( 1 + T + 89T^{2} \) |
| 97 | \( 1 + 2.93T + 97T^{2} \) |
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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
−7.49329799301234944739690438689, −6.90478896529621312141069860609, −5.99967411579618404485099634653, −5.36988903429113802946218590036, −4.76434793954581180660242228266, −3.56364745157746435020668959124, −2.90382709360988810352616176132, −2.21357485607849914560923789968, −1.04546294330198868463872502981, 0,
1.04546294330198868463872502981, 2.21357485607849914560923789968, 2.90382709360988810352616176132, 3.56364745157746435020668959124, 4.76434793954581180660242228266, 5.36988903429113802946218590036, 5.99967411579618404485099634653, 6.90478896529621312141069860609, 7.49329799301234944739690438689