| L(s) = 1 | + 2.12·3-s + 0.504·5-s + 1.53·9-s + 2.36·11-s − 1.21·13-s + 1.07·15-s − 3.03·17-s − 2.40·19-s − 23-s − 4.74·25-s − 3.12·27-s − 0.856·29-s − 4.69·31-s + 5.02·33-s − 11.9·37-s − 2.59·39-s + 9.46·41-s − 12.8·43-s + 0.772·45-s + 6.72·47-s − 6.46·51-s + 9.51·53-s + 1.19·55-s − 5.10·57-s + 6.44·59-s − 10.0·61-s − 0.615·65-s + ⋯ |
| L(s) = 1 | + 1.22·3-s + 0.225·5-s + 0.510·9-s + 0.711·11-s − 0.338·13-s + 0.277·15-s − 0.736·17-s − 0.550·19-s − 0.208·23-s − 0.949·25-s − 0.602·27-s − 0.158·29-s − 0.842·31-s + 0.874·33-s − 1.96·37-s − 0.415·39-s + 1.47·41-s − 1.95·43-s + 0.115·45-s + 0.980·47-s − 0.905·51-s + 1.30·53-s + 0.160·55-s − 0.676·57-s + 0.838·59-s − 1.28·61-s − 0.0763·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - 2.12T + 3T^{2} \) |
| 5 | \( 1 - 0.504T + 5T^{2} \) |
| 11 | \( 1 - 2.36T + 11T^{2} \) |
| 13 | \( 1 + 1.21T + 13T^{2} \) |
| 17 | \( 1 + 3.03T + 17T^{2} \) |
| 19 | \( 1 + 2.40T + 19T^{2} \) |
| 29 | \( 1 + 0.856T + 29T^{2} \) |
| 31 | \( 1 + 4.69T + 31T^{2} \) |
| 37 | \( 1 + 11.9T + 37T^{2} \) |
| 41 | \( 1 - 9.46T + 41T^{2} \) |
| 43 | \( 1 + 12.8T + 43T^{2} \) |
| 47 | \( 1 - 6.72T + 47T^{2} \) |
| 53 | \( 1 - 9.51T + 53T^{2} \) |
| 59 | \( 1 - 6.44T + 59T^{2} \) |
| 61 | \( 1 + 10.0T + 61T^{2} \) |
| 67 | \( 1 - 6.38T + 67T^{2} \) |
| 71 | \( 1 + 6.50T + 71T^{2} \) |
| 73 | \( 1 + 14.8T + 73T^{2} \) |
| 79 | \( 1 - 3.50T + 79T^{2} \) |
| 83 | \( 1 + 4.31T + 83T^{2} \) |
| 89 | \( 1 + 14.7T + 89T^{2} \) |
| 97 | \( 1 + 12.5T + 97T^{2} \) |
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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
−7.34257295212190856433819601537, −6.96577891363077648658317062117, −6.02680285587518608404677438991, −5.38256407517693207592753235913, −4.29436157166129818589658943554, −3.83840143362733336668574059635, −3.01614822034847185305320973906, −2.17178867522953761293604926624, −1.63735530268512235091303263358, 0,
1.63735530268512235091303263358, 2.17178867522953761293604926624, 3.01614822034847185305320973906, 3.83840143362733336668574059635, 4.29436157166129818589658943554, 5.38256407517693207592753235913, 6.02680285587518608404677438991, 6.96577891363077648658317062117, 7.34257295212190856433819601537
