| L(s) = 1 | + 3-s − 4.27·5-s − 7-s + 9-s + 2.27·11-s − 13-s − 4.27·15-s + 0.274·17-s + 2.27·19-s − 21-s − 2.27·23-s + 13.2·25-s + 27-s + 8.27·29-s − 8·31-s + 2.27·33-s + 4.27·35-s − 4.27·37-s − 39-s + 6.54·41-s + 2.27·43-s − 4.27·45-s + 49-s + 0.274·51-s + 10·53-s − 9.72·55-s + 2.27·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.91·5-s − 0.377·7-s + 0.333·9-s + 0.685·11-s − 0.277·13-s − 1.10·15-s + 0.0666·17-s + 0.521·19-s − 0.218·21-s − 0.474·23-s + 2.65·25-s + 0.192·27-s + 1.53·29-s − 1.43·31-s + 0.396·33-s + 0.722·35-s − 0.702·37-s − 0.160·39-s + 1.02·41-s + 0.346·43-s − 0.637·45-s + 0.142·49-s + 0.0384·51-s + 1.37·53-s − 1.31·55-s + 0.301·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + 4.27T + 5T^{2} \) |
| 11 | \( 1 - 2.27T + 11T^{2} \) |
| 17 | \( 1 - 0.274T + 17T^{2} \) |
| 19 | \( 1 - 2.27T + 19T^{2} \) |
| 23 | \( 1 + 2.27T + 23T^{2} \) |
| 29 | \( 1 - 8.27T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 4.27T + 37T^{2} \) |
| 41 | \( 1 - 6.54T + 41T^{2} \) |
| 43 | \( 1 - 2.27T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 10T + 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 + 12.5T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 12.8T + 73T^{2} \) |
| 79 | \( 1 + 12.5T + 79T^{2} \) |
| 83 | \( 1 - 4.54T + 83T^{2} \) |
| 89 | \( 1 + 14T + 89T^{2} \) |
| 97 | \( 1 - 15.0T + 97T^{2} \) |
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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
−7.889392026427067423410072936583, −7.40131674760284762396632088568, −6.87691475617728377043492692442, −5.85692606612607979901153052624, −4.65692390530241021993424807708, −4.13342882534706977989963386870, −3.41440056249570398766029959707, −2.77177238614580376965915239069, −1.26332050216584940928299380318, 0,
1.26332050216584940928299380318, 2.77177238614580376965915239069, 3.41440056249570398766029959707, 4.13342882534706977989963386870, 4.65692390530241021993424807708, 5.85692606612607979901153052624, 6.87691475617728377043492692442, 7.40131674760284762396632088568, 7.889392026427067423410072936583
