| L(s) = 1 | − 1.05·3-s + 3.24·5-s − 2.81·7-s − 1.87·9-s + 5.74·11-s − 2.81·13-s − 3.43·15-s − 5.80·17-s + 2.98·21-s + 4.37·23-s + 5.54·25-s + 5.16·27-s − 9.09·29-s − 5.54·31-s − 6.07·33-s − 9.14·35-s − 4.34·37-s + 2.98·39-s + 8.07·41-s − 0.0508·43-s − 6.10·45-s − 0.0700·47-s + 0.936·49-s + 6.14·51-s − 4.11·53-s + 18.6·55-s − 5.73·59-s + ⋯ |
| L(s) = 1 | − 0.611·3-s + 1.45·5-s − 1.06·7-s − 0.626·9-s + 1.73·11-s − 0.780·13-s − 0.887·15-s − 1.40·17-s + 0.650·21-s + 0.912·23-s + 1.10·25-s + 0.993·27-s − 1.68·29-s − 0.995·31-s − 1.05·33-s − 1.54·35-s − 0.714·37-s + 0.477·39-s + 1.26·41-s − 0.00775·43-s − 0.909·45-s − 0.0102·47-s + 0.133·49-s + 0.860·51-s − 0.564·53-s + 2.51·55-s − 0.746·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 1.05T + 3T^{2} \) |
| 5 | \( 1 - 3.24T + 5T^{2} \) |
| 7 | \( 1 + 2.81T + 7T^{2} \) |
| 11 | \( 1 - 5.74T + 11T^{2} \) |
| 13 | \( 1 + 2.81T + 13T^{2} \) |
| 17 | \( 1 + 5.80T + 17T^{2} \) |
| 23 | \( 1 - 4.37T + 23T^{2} \) |
| 29 | \( 1 + 9.09T + 29T^{2} \) |
| 31 | \( 1 + 5.54T + 31T^{2} \) |
| 37 | \( 1 + 4.34T + 37T^{2} \) |
| 41 | \( 1 - 8.07T + 41T^{2} \) |
| 43 | \( 1 + 0.0508T + 43T^{2} \) |
| 47 | \( 1 + 0.0700T + 47T^{2} \) |
| 53 | \( 1 + 4.11T + 53T^{2} \) |
| 59 | \( 1 + 5.73T + 59T^{2} \) |
| 61 | \( 1 + 3.98T + 61T^{2} \) |
| 67 | \( 1 + 8.45T + 67T^{2} \) |
| 71 | \( 1 + 9.37T + 71T^{2} \) |
| 73 | \( 1 + 3.12T + 73T^{2} \) |
| 79 | \( 1 - 9.24T + 79T^{2} \) |
| 83 | \( 1 - 13.7T + 83T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 97 | \( 1 - 13.7T + 97T^{2} \) |
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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
−8.867395682383846948577295050195, −7.29306149029029524233178670275, −6.61978483178241007634771506578, −6.15084613446287551148799907100, −5.54471622217782730206741809546, −4.61069992428309529658499966837, −3.51546310144781405176347967528, −2.50815074916430931020461402850, −1.54942872877654093203537952646, 0,
1.54942872877654093203537952646, 2.50815074916430931020461402850, 3.51546310144781405176347967528, 4.61069992428309529658499966837, 5.54471622217782730206741809546, 6.15084613446287551148799907100, 6.61978483178241007634771506578, 7.29306149029029524233178670275, 8.867395682383846948577295050195
