L(s) = 1 | − 1.79·3-s + 0.0213·5-s − 3.47·7-s + 0.236·9-s − 4.90·11-s − 0.505·13-s − 0.0384·15-s + 6.24·17-s + 5.27·19-s + 6.26·21-s + 3.59·23-s − 4.99·25-s + 4.97·27-s + 6.29·29-s − 0.796·31-s + 8.82·33-s − 0.0743·35-s + 4.25·37-s + 0.909·39-s + 9.96·41-s − 0.416·43-s + 0.00505·45-s + 0.618·47-s + 5.10·49-s − 11.2·51-s + 0.914·53-s − 0.104·55-s + ⋯ |
L(s) = 1 | − 1.03·3-s + 0.00955·5-s − 1.31·7-s + 0.0789·9-s − 1.47·11-s − 0.140·13-s − 0.00992·15-s + 1.51·17-s + 1.21·19-s + 1.36·21-s + 0.749·23-s − 0.999·25-s + 0.956·27-s + 1.16·29-s − 0.143·31-s + 1.53·33-s − 0.0125·35-s + 0.698·37-s + 0.145·39-s + 1.55·41-s − 0.0635·43-s + 0.000753·45-s + 0.0901·47-s + 0.729·49-s − 1.57·51-s + 0.125·53-s − 0.0141·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 + T \) |
good | 3 | \( 1 + 1.79T + 3T^{2} \) |
| 5 | \( 1 - 0.0213T + 5T^{2} \) |
| 7 | \( 1 + 3.47T + 7T^{2} \) |
| 11 | \( 1 + 4.90T + 11T^{2} \) |
| 13 | \( 1 + 0.505T + 13T^{2} \) |
| 17 | \( 1 - 6.24T + 17T^{2} \) |
| 19 | \( 1 - 5.27T + 19T^{2} \) |
| 23 | \( 1 - 3.59T + 23T^{2} \) |
| 29 | \( 1 - 6.29T + 29T^{2} \) |
| 31 | \( 1 + 0.796T + 31T^{2} \) |
| 37 | \( 1 - 4.25T + 37T^{2} \) |
| 41 | \( 1 - 9.96T + 41T^{2} \) |
| 43 | \( 1 + 0.416T + 43T^{2} \) |
| 47 | \( 1 - 0.618T + 47T^{2} \) |
| 53 | \( 1 - 0.914T + 53T^{2} \) |
| 59 | \( 1 + 10.6T + 59T^{2} \) |
| 61 | \( 1 + 7.61T + 61T^{2} \) |
| 67 | \( 1 - 3.27T + 67T^{2} \) |
| 71 | \( 1 + 12.8T + 71T^{2} \) |
| 73 | \( 1 + 12.7T + 73T^{2} \) |
| 79 | \( 1 + 0.878T + 79T^{2} \) |
| 83 | \( 1 + 12.0T + 83T^{2} \) |
| 89 | \( 1 - 12.6T + 89T^{2} \) |
| 97 | \( 1 + 5.12T + 97T^{2} \) |
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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
−7.76688725190240231199893225860, −7.44589585548358782599020890797, −6.39315154229704228393660218946, −5.76270526892571359202672885224, −5.37316561544538856633625566179, −4.44906752088531072435704282768, −3.10167136926549359086662836572, −2.85163769647703526078969638703, −1.03335843446519394222110130035, 0,
1.03335843446519394222110130035, 2.85163769647703526078969638703, 3.10167136926549359086662836572, 4.44906752088531072435704282768, 5.37316561544538856633625566179, 5.76270526892571359202672885224, 6.39315154229704228393660218946, 7.44589585548358782599020890797, 7.76688725190240231199893225860