| L(s) = 1 | + 1.41·2-s + 2.44·7-s − 2.82·8-s + 4.73·11-s + 2.44·13-s + 3.46·14-s − 4.00·16-s + 2.96·17-s + 3.19·19-s + 6.69·22-s − 1.41·23-s + 3.46·26-s − 2.19·29-s + 31-s + 4.19·34-s + 0.896·37-s + 4.52·38-s + 5.53·41-s − 12.4·43-s − 2.00·46-s + 8.76·47-s − 1.00·49-s + 1.27·53-s − 6.92·56-s + ⋯ |
| L(s) = 1 | + 1.00·2-s + 0.925·7-s − 0.999·8-s + 1.42·11-s + 0.679·13-s + 0.925·14-s − 1.00·16-s + 0.719·17-s + 0.733·19-s + 1.42·22-s − 0.294·23-s + 0.679·26-s − 0.407·29-s + 0.179·31-s + 0.719·34-s + 0.147·37-s + 0.733·38-s + 0.864·41-s − 1.90·43-s − 0.294·46-s + 1.27·47-s − 0.142·49-s + 0.175·53-s − 0.925·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.968635265\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.968635265\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 - 1.41T + 2T^{2} \) |
| 7 | \( 1 - 2.44T + 7T^{2} \) |
| 11 | \( 1 - 4.73T + 11T^{2} \) |
| 13 | \( 1 - 2.44T + 13T^{2} \) |
| 17 | \( 1 - 2.96T + 17T^{2} \) |
| 19 | \( 1 - 3.19T + 19T^{2} \) |
| 23 | \( 1 + 1.41T + 23T^{2} \) |
| 29 | \( 1 + 2.19T + 29T^{2} \) |
| 37 | \( 1 - 0.896T + 37T^{2} \) |
| 41 | \( 1 - 5.53T + 41T^{2} \) |
| 43 | \( 1 + 12.4T + 43T^{2} \) |
| 47 | \( 1 - 8.76T + 47T^{2} \) |
| 53 | \( 1 - 1.27T + 53T^{2} \) |
| 59 | \( 1 - 9.92T + 59T^{2} \) |
| 61 | \( 1 + 14.3T + 61T^{2} \) |
| 67 | \( 1 + 2.44T + 67T^{2} \) |
| 71 | \( 1 - 7.73T + 71T^{2} \) |
| 73 | \( 1 - 8.24T + 73T^{2} \) |
| 79 | \( 1 + 4.19T + 79T^{2} \) |
| 83 | \( 1 + 14.5T + 83T^{2} \) |
| 89 | \( 1 - 14.1T + 89T^{2} \) |
| 97 | \( 1 - 11.5T + 97T^{2} \) |
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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
−7.981255720495153106526910915271, −7.10502490281669169674659303195, −6.32870115968779479981142346173, −5.71829238311320065156863630113, −5.07680415465888400548164366296, −4.30431329752145595533834103340, −3.75517625689667933686147911631, −3.06652463434069415940418923352, −1.82292988114402776087063892181, −0.939973024091356966375057473329,
0.939973024091356966375057473329, 1.82292988114402776087063892181, 3.06652463434069415940418923352, 3.75517625689667933686147911631, 4.30431329752145595533834103340, 5.07680415465888400548164366296, 5.71829238311320065156863630113, 6.32870115968779479981142346173, 7.10502490281669169674659303195, 7.981255720495153106526910915271
