L(s) = 1 | + 2-s + 2.18·3-s + 4-s − 0.616·5-s + 2.18·6-s − 7-s + 8-s + 1.78·9-s − 0.616·10-s + 3.42·11-s + 2.18·12-s + 6.38·13-s − 14-s − 1.34·15-s + 16-s + 2.47·17-s + 1.78·18-s − 0.616·20-s − 2.18·21-s + 3.42·22-s + 2.70·23-s + 2.18·24-s − 4.62·25-s + 6.38·26-s − 2.65·27-s − 28-s + 6.00·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.26·3-s + 0.5·4-s − 0.275·5-s + 0.893·6-s − 0.377·7-s + 0.353·8-s + 0.595·9-s − 0.194·10-s + 1.03·11-s + 0.631·12-s + 1.76·13-s − 0.267·14-s − 0.348·15-s + 0.250·16-s + 0.601·17-s + 0.420·18-s − 0.137·20-s − 0.477·21-s + 0.731·22-s + 0.564·23-s + 0.446·24-s − 0.924·25-s + 1.25·26-s − 0.511·27-s − 0.188·28-s + 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.325260823\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.325260823\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 - 2.18T + 3T^{2} \) |
| 5 | \( 1 + 0.616T + 5T^{2} \) |
| 11 | \( 1 - 3.42T + 11T^{2} \) |
| 13 | \( 1 - 6.38T + 13T^{2} \) |
| 17 | \( 1 - 2.47T + 17T^{2} \) |
| 23 | \( 1 - 2.70T + 23T^{2} \) |
| 29 | \( 1 - 6.00T + 29T^{2} \) |
| 31 | \( 1 + 7.72T + 31T^{2} \) |
| 37 | \( 1 - 4.26T + 37T^{2} \) |
| 41 | \( 1 + 1.28T + 41T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 47 | \( 1 - 1.53T + 47T^{2} \) |
| 53 | \( 1 + 5.11T + 53T^{2} \) |
| 59 | \( 1 - 12.7T + 59T^{2} \) |
| 61 | \( 1 + 5.54T + 61T^{2} \) |
| 67 | \( 1 - 4.16T + 67T^{2} \) |
| 71 | \( 1 - 0.0244T + 71T^{2} \) |
| 73 | \( 1 - 13.0T + 73T^{2} \) |
| 79 | \( 1 - 2.96T + 79T^{2} \) |
| 83 | \( 1 - 12.9T + 83T^{2} \) |
| 89 | \( 1 + 12.4T + 89T^{2} \) |
| 97 | \( 1 + 1.51T + 97T^{2} \) |
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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
−8.284835843164048190655063147187, −7.59152254830742051043580699108, −6.69607831708006092332503626480, −6.17330220188858657186470070759, −5.30011127359153927836015592082, −4.14709452540210573984728176305, −3.57118926098274063779054625006, −3.25120347078235225972981460407, −2.09444808907057660496203354794, −1.16807746960069608293794963456,
1.16807746960069608293794963456, 2.09444808907057660496203354794, 3.25120347078235225972981460407, 3.57118926098274063779054625006, 4.14709452540210573984728176305, 5.30011127359153927836015592082, 6.17330220188858657186470070759, 6.69607831708006092332503626480, 7.59152254830742051043580699108, 8.284835843164048190655063147187