| L(s) = 1 | + 2.62·2-s − 0.427·3-s + 4.87·4-s − 2.35·5-s − 1.12·6-s − 7-s + 7.55·8-s − 2.81·9-s − 6.17·10-s − 4.75·11-s − 2.08·12-s − 2.29·13-s − 2.62·14-s + 1.00·15-s + 10.0·16-s − 7.38·18-s − 6.15·19-s − 11.4·20-s + 0.427·21-s − 12.4·22-s + 1.67·23-s − 3.22·24-s + 0.547·25-s − 6.01·26-s + 2.48·27-s − 4.87·28-s + 0.385·29-s + ⋯ |
| L(s) = 1 | + 1.85·2-s − 0.246·3-s + 2.43·4-s − 1.05·5-s − 0.457·6-s − 0.377·7-s + 2.66·8-s − 0.939·9-s − 1.95·10-s − 1.43·11-s − 0.601·12-s − 0.636·13-s − 0.700·14-s + 0.259·15-s + 2.51·16-s − 1.74·18-s − 1.41·19-s − 2.56·20-s + 0.0932·21-s − 2.66·22-s + 0.349·23-s − 0.658·24-s + 0.109·25-s − 1.18·26-s + 0.478·27-s − 0.922·28-s + 0.0716·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 2.62T + 2T^{2} \) |
| 3 | \( 1 + 0.427T + 3T^{2} \) |
| 5 | \( 1 + 2.35T + 5T^{2} \) |
| 11 | \( 1 + 4.75T + 11T^{2} \) |
| 13 | \( 1 + 2.29T + 13T^{2} \) |
| 19 | \( 1 + 6.15T + 19T^{2} \) |
| 23 | \( 1 - 1.67T + 23T^{2} \) |
| 29 | \( 1 - 0.385T + 29T^{2} \) |
| 31 | \( 1 - 2.16T + 31T^{2} \) |
| 37 | \( 1 - 8.49T + 37T^{2} \) |
| 41 | \( 1 + 6.56T + 41T^{2} \) |
| 43 | \( 1 - 5.54T + 43T^{2} \) |
| 47 | \( 1 + 12.5T + 47T^{2} \) |
| 53 | \( 1 + 0.942T + 53T^{2} \) |
| 59 | \( 1 - 2.18T + 59T^{2} \) |
| 61 | \( 1 + 8.41T + 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 - 1.46T + 71T^{2} \) |
| 73 | \( 1 - 13.9T + 73T^{2} \) |
| 79 | \( 1 + 4.80T + 79T^{2} \) |
| 83 | \( 1 - 6.31T + 83T^{2} \) |
| 89 | \( 1 + 6.22T + 89T^{2} \) |
| 97 | \( 1 + 11.7T + 97T^{2} \) |
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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
−8.293861502485300484209293731291, −7.84706873367060276218941928204, −6.86124487047123379874294021976, −6.20147716785571944884693606187, −5.31621570343207376424318837447, −4.73600331482578481202548833749, −3.89280634320072440644688702350, −2.96491401132770587673019337644, −2.37570063440469333274322461503, 0,
2.37570063440469333274322461503, 2.96491401132770587673019337644, 3.89280634320072440644688702350, 4.73600331482578481202548833749, 5.31621570343207376424318837447, 6.20147716785571944884693606187, 6.86124487047123379874294021976, 7.84706873367060276218941928204, 8.293861502485300484209293731291
