| L(s) = 1 | − 1.76·3-s − 5-s − 3.89·7-s + 0.114·9-s + 4.80·11-s + 0.0353·13-s + 1.76·15-s − 4.39·17-s + 19-s + 6.86·21-s + 9.12·23-s + 25-s + 5.09·27-s − 6.21·29-s + 9.71·31-s − 8.48·33-s + 3.89·35-s − 0.749·37-s − 0.0623·39-s − 3.52·41-s − 11.4·43-s − 0.114·45-s + 5.57·47-s + 8.15·49-s + 7.75·51-s + 8.67·53-s − 4.80·55-s + ⋯ |
| L(s) = 1 | − 1.01·3-s − 0.447·5-s − 1.47·7-s + 0.0382·9-s + 1.44·11-s + 0.00979·13-s + 0.455·15-s − 1.06·17-s + 0.229·19-s + 1.49·21-s + 1.90·23-s + 0.200·25-s + 0.979·27-s − 1.15·29-s + 1.74·31-s − 1.47·33-s + 0.657·35-s − 0.123·37-s − 0.00997·39-s − 0.551·41-s − 1.74·43-s − 0.0171·45-s + 0.813·47-s + 1.16·49-s + 1.08·51-s + 1.19·53-s − 0.648·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 1.76T + 3T^{2} \) |
| 7 | \( 1 + 3.89T + 7T^{2} \) |
| 11 | \( 1 - 4.80T + 11T^{2} \) |
| 13 | \( 1 - 0.0353T + 13T^{2} \) |
| 17 | \( 1 + 4.39T + 17T^{2} \) |
| 23 | \( 1 - 9.12T + 23T^{2} \) |
| 29 | \( 1 + 6.21T + 29T^{2} \) |
| 31 | \( 1 - 9.71T + 31T^{2} \) |
| 37 | \( 1 + 0.749T + 37T^{2} \) |
| 41 | \( 1 + 3.52T + 41T^{2} \) |
| 43 | \( 1 + 11.4T + 43T^{2} \) |
| 47 | \( 1 - 5.57T + 47T^{2} \) |
| 53 | \( 1 - 8.67T + 53T^{2} \) |
| 59 | \( 1 + 2.56T + 59T^{2} \) |
| 61 | \( 1 - 6.50T + 61T^{2} \) |
| 67 | \( 1 + 13.2T + 67T^{2} \) |
| 71 | \( 1 - 1.67T + 71T^{2} \) |
| 73 | \( 1 + 11.3T + 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 + 5.64T + 83T^{2} \) |
| 89 | \( 1 + 1.24T + 89T^{2} \) |
| 97 | \( 1 - 9.31T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.614307465922845809285575539878, −7.19458956713921550823211956239, −6.67217153530042521241289988350, −6.29559683470374880598697769063, −5.33706283337834083658561354474, −4.45814702933879927704555817115, −3.58515212804073181565946697371, −2.78558076696772335986770746362, −1.11230830272604284563183722380, 0,
1.11230830272604284563183722380, 2.78558076696772335986770746362, 3.58515212804073181565946697371, 4.45814702933879927704555817115, 5.33706283337834083658561354474, 6.29559683470374880598697769063, 6.67217153530042521241289988350, 7.19458956713921550823211956239, 8.614307465922845809285575539878
