| L(s) = 1 | − 2.46·2-s + 0.936·3-s + 4.08·4-s + 5-s − 2.31·6-s + 7-s − 5.13·8-s − 2.12·9-s − 2.46·10-s − 4.45·11-s + 3.82·12-s − 1.89·13-s − 2.46·14-s + 0.936·15-s + 4.49·16-s + 1.76·17-s + 5.23·18-s + 4.27·19-s + 4.08·20-s + 0.936·21-s + 10.9·22-s + 2.15·23-s − 4.81·24-s + 25-s + 4.68·26-s − 4.79·27-s + 4.08·28-s + ⋯ |
| L(s) = 1 | − 1.74·2-s + 0.540·3-s + 2.04·4-s + 0.447·5-s − 0.943·6-s + 0.377·7-s − 1.81·8-s − 0.707·9-s − 0.779·10-s − 1.34·11-s + 1.10·12-s − 0.526·13-s − 0.659·14-s + 0.241·15-s + 1.12·16-s + 0.426·17-s + 1.23·18-s + 0.981·19-s + 0.912·20-s + 0.204·21-s + 2.34·22-s + 0.449·23-s − 0.981·24-s + 0.200·25-s + 0.917·26-s − 0.923·27-s + 0.771·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 229 | \( 1 + T \) |
| good | 2 | \( 1 + 2.46T + 2T^{2} \) |
| 3 | \( 1 - 0.936T + 3T^{2} \) |
| 11 | \( 1 + 4.45T + 11T^{2} \) |
| 13 | \( 1 + 1.89T + 13T^{2} \) |
| 17 | \( 1 - 1.76T + 17T^{2} \) |
| 19 | \( 1 - 4.27T + 19T^{2} \) |
| 23 | \( 1 - 2.15T + 23T^{2} \) |
| 29 | \( 1 - 4.26T + 29T^{2} \) |
| 31 | \( 1 - 1.97T + 31T^{2} \) |
| 37 | \( 1 + 8.92T + 37T^{2} \) |
| 41 | \( 1 + 2.93T + 41T^{2} \) |
| 43 | \( 1 + 8.72T + 43T^{2} \) |
| 47 | \( 1 - 8.43T + 47T^{2} \) |
| 53 | \( 1 + 1.44T + 53T^{2} \) |
| 59 | \( 1 + 0.429T + 59T^{2} \) |
| 61 | \( 1 + 0.159T + 61T^{2} \) |
| 67 | \( 1 - 6.32T + 67T^{2} \) |
| 71 | \( 1 - 4.09T + 71T^{2} \) |
| 73 | \( 1 - 1.94T + 73T^{2} \) |
| 79 | \( 1 - 1.65T + 79T^{2} \) |
| 83 | \( 1 - 14.7T + 83T^{2} \) |
| 89 | \( 1 - 7.26T + 89T^{2} \) |
| 97 | \( 1 + 2.28T + 97T^{2} \) |
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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
−7.77134269883671297849156558083, −7.16885819198292793334212592325, −6.42188521785871974854332236095, −5.38004223766748399571729279994, −5.02686290304609421292701789097, −3.41215666819389127551952423270, −2.69089478554710197666901307669, −2.15172127317885833910362224084, −1.13097958510945351349415693333, 0,
1.13097958510945351349415693333, 2.15172127317885833910362224084, 2.69089478554710197666901307669, 3.41215666819389127551952423270, 5.02686290304609421292701789097, 5.38004223766748399571729279994, 6.42188521785871974854332236095, 7.16885819198292793334212592325, 7.77134269883671297849156558083
