L(s) = 1 | − 2-s − 1.97·3-s + 4-s − 3.11·5-s + 1.97·6-s − 3.94·7-s − 8-s + 0.909·9-s + 3.11·10-s − 4.52·11-s − 1.97·12-s − 5.66·13-s + 3.94·14-s + 6.16·15-s + 16-s + 1.59·17-s − 0.909·18-s − 6.00·19-s − 3.11·20-s + 7.79·21-s + 4.52·22-s + 23-s + 1.97·24-s + 4.71·25-s + 5.66·26-s + 4.13·27-s − 3.94·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.14·3-s + 0.5·4-s − 1.39·5-s + 0.807·6-s − 1.49·7-s − 0.353·8-s + 0.303·9-s + 0.985·10-s − 1.36·11-s − 0.570·12-s − 1.57·13-s + 1.05·14-s + 1.59·15-s + 0.250·16-s + 0.386·17-s − 0.214·18-s − 1.37·19-s − 0.696·20-s + 1.70·21-s + 0.964·22-s + 0.208·23-s + 0.403·24-s + 0.942·25-s + 1.11·26-s + 0.795·27-s − 0.745·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 - T \) |
good | 3 | \( 1 + 1.97T + 3T^{2} \) |
| 5 | \( 1 + 3.11T + 5T^{2} \) |
| 7 | \( 1 + 3.94T + 7T^{2} \) |
| 11 | \( 1 + 4.52T + 11T^{2} \) |
| 13 | \( 1 + 5.66T + 13T^{2} \) |
| 17 | \( 1 - 1.59T + 17T^{2} \) |
| 19 | \( 1 + 6.00T + 19T^{2} \) |
| 29 | \( 1 + 5.05T + 29T^{2} \) |
| 31 | \( 1 - 3.72T + 31T^{2} \) |
| 37 | \( 1 - 3.72T + 37T^{2} \) |
| 41 | \( 1 - 2.67T + 41T^{2} \) |
| 43 | \( 1 - 6.70T + 43T^{2} \) |
| 47 | \( 1 + 2.96T + 47T^{2} \) |
| 53 | \( 1 + 2.04T + 53T^{2} \) |
| 59 | \( 1 - 1.88T + 59T^{2} \) |
| 61 | \( 1 - 2.55T + 61T^{2} \) |
| 67 | \( 1 - 4.61T + 67T^{2} \) |
| 71 | \( 1 - 5.59T + 71T^{2} \) |
| 73 | \( 1 + 13.5T + 73T^{2} \) |
| 79 | \( 1 + 14.9T + 79T^{2} \) |
| 83 | \( 1 + 9.80T + 83T^{2} \) |
| 89 | \( 1 + 0.355T + 89T^{2} \) |
| 97 | \( 1 - 14.7T + 97T^{2} \) |
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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
−7.56136429929424482212172067202, −7.19044524745309832956818213793, −6.37787263997042374985095215259, −5.73902230440049315361806823105, −4.89420527043196454694714407779, −4.11884589395749143249985460794, −3.04654943877487540651907448684, −2.46259647805388547266498770300, −0.50940447870168803271480564601, 0,
0.50940447870168803271480564601, 2.46259647805388547266498770300, 3.04654943877487540651907448684, 4.11884589395749143249985460794, 4.89420527043196454694714407779, 5.73902230440049315361806823105, 6.37787263997042374985095215259, 7.19044524745309832956818213793, 7.56136429929424482212172067202