| L(s) = 1 | − 0.682·3-s + 3.69·5-s + 3.46·7-s − 2.53·9-s − 1.14·11-s − 0.915·13-s − 2.52·15-s − 7.09·17-s + 3.54·19-s − 2.36·21-s − 4.97·23-s + 8.65·25-s + 3.77·27-s − 4.41·29-s − 10.1·31-s + 0.781·33-s + 12.8·35-s + 2.60·37-s + 0.624·39-s − 9.41·41-s − 0.391·43-s − 9.36·45-s + 4.45·47-s + 5.01·49-s + 4.84·51-s − 1.57·53-s − 4.23·55-s + ⋯ |
| L(s) = 1 | − 0.393·3-s + 1.65·5-s + 1.31·7-s − 0.844·9-s − 0.345·11-s − 0.254·13-s − 0.651·15-s − 1.72·17-s + 0.812·19-s − 0.516·21-s − 1.03·23-s + 1.73·25-s + 0.726·27-s − 0.820·29-s − 1.82·31-s + 0.136·33-s + 2.16·35-s + 0.428·37-s + 0.100·39-s − 1.47·41-s − 0.0597·43-s − 1.39·45-s + 0.650·47-s + 0.716·49-s + 0.678·51-s − 0.216·53-s − 0.570·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 0.682T + 3T^{2} \) |
| 5 | \( 1 - 3.69T + 5T^{2} \) |
| 7 | \( 1 - 3.46T + 7T^{2} \) |
| 11 | \( 1 + 1.14T + 11T^{2} \) |
| 13 | \( 1 + 0.915T + 13T^{2} \) |
| 17 | \( 1 + 7.09T + 17T^{2} \) |
| 19 | \( 1 - 3.54T + 19T^{2} \) |
| 23 | \( 1 + 4.97T + 23T^{2} \) |
| 29 | \( 1 + 4.41T + 29T^{2} \) |
| 31 | \( 1 + 10.1T + 31T^{2} \) |
| 37 | \( 1 - 2.60T + 37T^{2} \) |
| 41 | \( 1 + 9.41T + 41T^{2} \) |
| 43 | \( 1 + 0.391T + 43T^{2} \) |
| 47 | \( 1 - 4.45T + 47T^{2} \) |
| 53 | \( 1 + 1.57T + 53T^{2} \) |
| 59 | \( 1 - 3.21T + 59T^{2} \) |
| 61 | \( 1 + 5.61T + 61T^{2} \) |
| 67 | \( 1 - 0.0216T + 67T^{2} \) |
| 71 | \( 1 - 1.49T + 71T^{2} \) |
| 73 | \( 1 + 12.8T + 73T^{2} \) |
| 79 | \( 1 - 6.15T + 79T^{2} \) |
| 83 | \( 1 - 0.0862T + 83T^{2} \) |
| 89 | \( 1 + 16.8T + 89T^{2} \) |
| 97 | \( 1 + 10.8T + 97T^{2} \) |
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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
−7.45264625743824337092539057774, −6.65984526807490432177143609124, −5.93251833960092697747366735462, −5.36536036270382640424699920350, −5.02931200821372670632842920327, −4.06741482574898590969798930259, −2.80261700259308324972320402381, −2.05437066907073049155435535811, −1.56947505273624099826573715630, 0,
1.56947505273624099826573715630, 2.05437066907073049155435535811, 2.80261700259308324972320402381, 4.06741482574898590969798930259, 5.02931200821372670632842920327, 5.36536036270382640424699920350, 5.93251833960092697747366735462, 6.65984526807490432177143609124, 7.45264625743824337092539057774
