| L(s) = 1 | − 2-s − 2.69·3-s + 4-s − 3.24·5-s + 2.69·6-s + 3.60·7-s − 8-s + 4.24·9-s + 3.24·10-s + 5.15·11-s − 2.69·12-s − 3.60·14-s + 8.74·15-s + 16-s + 1.40·17-s − 4.24·18-s + 19-s − 3.24·20-s − 9.70·21-s − 5.15·22-s + 5.43·23-s + 2.69·24-s + 5.54·25-s − 3.35·27-s + 3.60·28-s − 5.15·29-s − 8.74·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.55·3-s + 0.5·4-s − 1.45·5-s + 1.09·6-s + 1.36·7-s − 0.353·8-s + 1.41·9-s + 1.02·10-s + 1.55·11-s − 0.777·12-s − 0.963·14-s + 2.25·15-s + 0.250·16-s + 0.340·17-s − 1.00·18-s + 0.229·19-s − 0.726·20-s − 2.11·21-s − 1.09·22-s + 1.13·23-s + 0.549·24-s + 1.10·25-s − 0.646·27-s + 0.681·28-s − 0.957·29-s − 1.59·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 2.69T + 3T^{2} \) |
| 5 | \( 1 + 3.24T + 5T^{2} \) |
| 7 | \( 1 - 3.60T + 7T^{2} \) |
| 11 | \( 1 - 5.15T + 11T^{2} \) |
| 17 | \( 1 - 1.40T + 17T^{2} \) |
| 23 | \( 1 - 5.43T + 23T^{2} \) |
| 29 | \( 1 + 5.15T + 29T^{2} \) |
| 31 | \( 1 + 6.85T + 31T^{2} \) |
| 37 | \( 1 + 9.42T + 37T^{2} \) |
| 41 | \( 1 - 2.59T + 41T^{2} \) |
| 43 | \( 1 + 12.7T + 43T^{2} \) |
| 47 | \( 1 + 6.71T + 47T^{2} \) |
| 53 | \( 1 - 5.74T + 53T^{2} \) |
| 59 | \( 1 - 4.98T + 59T^{2} \) |
| 61 | \( 1 - 9.08T + 61T^{2} \) |
| 67 | \( 1 - 0.176T + 67T^{2} \) |
| 71 | \( 1 + 5.07T + 71T^{2} \) |
| 73 | \( 1 - 9.92T + 73T^{2} \) |
| 79 | \( 1 + 17.0T + 79T^{2} \) |
| 83 | \( 1 - 5.70T + 83T^{2} \) |
| 89 | \( 1 - 11.6T + 89T^{2} \) |
| 97 | \( 1 - 7.75T + 97T^{2} \) |
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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
−7.50233771019129776929901057540, −7.06487851466018646515211569762, −6.49745312065837093610665096407, −5.36766965513995740392672831105, −5.01790754967283537599887272828, −4.05059297450507927536238143209, −3.48613426784102671258838923961, −1.70383999671082671329719805779, −1.06871188386316944253574265178, 0,
1.06871188386316944253574265178, 1.70383999671082671329719805779, 3.48613426784102671258838923961, 4.05059297450507927536238143209, 5.01790754967283537599887272828, 5.36766965513995740392672831105, 6.49745312065837093610665096407, 7.06487851466018646515211569762, 7.50233771019129776929901057540
