| L(s) = 1 | − 1.41·2-s + 2.64·5-s + 7-s + 2.82·8-s − 3.74·10-s − 5.74·13-s − 1.41·14-s − 4.00·16-s − 5.47·17-s + 5.74·19-s − 3.87·23-s + 2.00·25-s + 8.11·26-s − 3.87·29-s + 5.48·31-s + 7.74·34-s + 2.64·35-s + 3.48·37-s − 8.11·38-s + 7.48·40-s + 5.65·41-s − 43-s + 5.48·46-s − 5.47·47-s + 49-s + ⋯ |
| L(s) = 1 | − 1.00·2-s + 1.18·5-s + 0.377·7-s + 0.999·8-s − 1.18·10-s − 1.59·13-s − 0.377·14-s − 1.00·16-s − 1.32·17-s + 1.31·19-s − 0.808·23-s + 0.400·25-s + 1.59·26-s − 0.719·29-s + 0.984·31-s + 1.32·34-s + 0.447·35-s + 0.572·37-s − 1.31·38-s + 1.18·40-s + 0.883·41-s − 0.152·43-s + 0.808·46-s − 0.798·47-s + 0.142·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 1.41T + 2T^{2} \) |
| 5 | \( 1 - 2.64T + 5T^{2} \) |
| 13 | \( 1 + 5.74T + 13T^{2} \) |
| 17 | \( 1 + 5.47T + 17T^{2} \) |
| 19 | \( 1 - 5.74T + 19T^{2} \) |
| 23 | \( 1 + 3.87T + 23T^{2} \) |
| 29 | \( 1 + 3.87T + 29T^{2} \) |
| 31 | \( 1 - 5.48T + 31T^{2} \) |
| 37 | \( 1 - 3.48T + 37T^{2} \) |
| 41 | \( 1 - 5.65T + 41T^{2} \) |
| 43 | \( 1 + T + 43T^{2} \) |
| 47 | \( 1 + 5.47T + 47T^{2} \) |
| 53 | \( 1 - 13.4T + 53T^{2} \) |
| 59 | \( 1 - 8.30T + 59T^{2} \) |
| 61 | \( 1 - 1.74T + 61T^{2} \) |
| 67 | \( 1 + 6.48T + 67T^{2} \) |
| 71 | \( 1 + 1.41T + 71T^{2} \) |
| 73 | \( 1 + 15.7T + 73T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 + 11.1T + 83T^{2} \) |
| 89 | \( 1 - 5.83T + 89T^{2} \) |
| 97 | \( 1 + 7.22T + 97T^{2} \) |
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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
−7.42667021264376589448395659785, −7.22833906197895020322562685263, −6.16267718121126874752055959284, −5.44149297265585979603581027937, −4.76632000089767777116205523627, −4.12274533110793621863411326408, −2.64797754110777421533853171006, −2.11718184544033026087738332169, −1.22197095139936685855159162192, 0,
1.22197095139936685855159162192, 2.11718184544033026087738332169, 2.64797754110777421533853171006, 4.12274533110793621863411326408, 4.76632000089767777116205523627, 5.44149297265585979603581027937, 6.16267718121126874752055959284, 7.22833906197895020322562685263, 7.42667021264376589448395659785
