L(s) = 1 | + 2-s − 3-s − 4-s − 2·5-s − 6-s − 7-s − 3·8-s + 9-s − 2·10-s + 4·11-s + 12-s + 2·13-s − 14-s + 2·15-s − 16-s − 6·17-s + 18-s + 2·20-s + 21-s + 4·22-s + 3·24-s − 25-s + 2·26-s − 27-s + 28-s + 2·29-s + 2·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.894·5-s − 0.408·6-s − 0.377·7-s − 1.06·8-s + 1/3·9-s − 0.632·10-s + 1.20·11-s + 0.288·12-s + 0.554·13-s − 0.267·14-s + 0.516·15-s − 1/4·16-s − 1.45·17-s + 0.235·18-s + 0.447·20-s + 0.218·21-s + 0.852·22-s + 0.612·24-s − 1/5·25-s + 0.392·26-s − 0.192·27-s + 0.188·28-s + 0.371·29-s + 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7581 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7581 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8768222459\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8768222459\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−7.85302053965317674139971119169, −6.87477517530779045173647189781, −6.42842960047713503559988653252, −5.82904766004805566557712895862, −4.81936947701198622186082780840, −4.36029050874640221146718646433, −3.73298077731310404370734463266, −3.13006637509653258681236844743, −1.71320545529150591858508044629, −0.43525041789359895883868375518,
0.43525041789359895883868375518, 1.71320545529150591858508044629, 3.13006637509653258681236844743, 3.73298077731310404370734463266, 4.36029050874640221146718646433, 4.81936947701198622186082780840, 5.82904766004805566557712895862, 6.42842960047713503559988653252, 6.87477517530779045173647189781, 7.85302053965317674139971119169