| L(s) = 1 | + 2-s + 4-s + 7-s + 8-s + 5.29·11-s − 4.64·13-s + 14-s + 16-s − 17-s − 6.29·19-s + 5.29·22-s + 1.64·23-s − 4.64·26-s + 28-s − 10.6·29-s − 7.29·31-s + 32-s − 34-s − 9.64·37-s − 6.29·38-s + 7.29·41-s + 0.354·43-s + 5.29·44-s + 1.64·46-s + 0.645·47-s + 49-s − 4.64·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.377·7-s + 0.353·8-s + 1.59·11-s − 1.28·13-s + 0.267·14-s + 0.250·16-s − 0.242·17-s − 1.44·19-s + 1.12·22-s + 0.343·23-s − 0.911·26-s + 0.188·28-s − 1.97·29-s − 1.30·31-s + 0.176·32-s − 0.171·34-s − 1.58·37-s − 1.02·38-s + 1.13·41-s + 0.0540·43-s + 0.797·44-s + 0.242·46-s + 0.0941·47-s + 0.142·49-s − 0.644·52-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 | \( 1 + 4.64T + 13T^{2} \) |
| 17 | \( 1 + T + 17T^{2} \) |
| 19 | \( 1 + 6.29T + 19T^{2} \) |
| 23 | \( 1 - 1.64T + 23T^{2} \) |
| 29 | \( 1 + 10.6T + 29T^{2} \) |
| 31 | \( 1 + 7.29T + 31T^{2} \) |
| 37 | \( 1 + 9.64T + 37T^{2} \) |
| 41 | \( 1 - 7.29T + 41T^{2} \) |
| 43 | \( 1 - 0.354T + 43T^{2} \) |
| 47 | \( 1 - 0.645T + 47T^{2} \) |
| 53 | \( 1 - 2.64T + 53T^{2} \) |
| 59 | \( 1 + 13.2T + 59T^{2} \) |
| 61 | \( 1 - 2.64T + 61T^{2} \) |
| 67 | \( 1 - 4.93T + 67T^{2} \) |
| 71 | \( 1 + 11.2T + 71T^{2} \) |
| 73 | \( 1 + 5.29T + 73T^{2} \) |
| 79 | \( 1 + 15.9T + 79T^{2} \) |
| 83 | \( 1 - 3.64T + 83T^{2} \) |
| 89 | \( 1 + T + 89T^{2} \) |
| 97 | \( 1 + 12.9T + 97T^{2} \) |
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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
−7.17775772470996343677400210075, −6.74277471250511831987116303323, −5.87804760691468130619418759773, −5.32273791973839887305771779361, −4.36645308915058738366648109579, −4.06979752712028210548095356683, −3.16427612826954578984477228955, −2.09693604735788206047005962121, −1.59763979488773294201109868093, 0,
1.59763979488773294201109868093, 2.09693604735788206047005962121, 3.16427612826954578984477228955, 4.06979752712028210548095356683, 4.36645308915058738366648109579, 5.32273791973839887305771779361, 5.87804760691468130619418759773, 6.74277471250511831987116303323, 7.17775772470996343677400210075
