L(s) = 1 | + 2-s + 4-s + 2.16·5-s + 4.36·7-s + 8-s + 2.16·10-s + 3.92·11-s − 0.954·13-s + 4.36·14-s + 16-s − 2.48·17-s + 2.16·20-s + 3.92·22-s + 7.07·23-s − 0.304·25-s − 0.954·26-s + 4.36·28-s − 9.55·29-s + 1.50·31-s + 32-s − 2.48·34-s + 9.44·35-s + 6.81·37-s + 2.16·40-s + 0.523·41-s − 9.04·43-s + 3.92·44-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.969·5-s + 1.64·7-s + 0.353·8-s + 0.685·10-s + 1.18·11-s − 0.264·13-s + 1.16·14-s + 0.250·16-s − 0.603·17-s + 0.484·20-s + 0.836·22-s + 1.47·23-s − 0.0608·25-s − 0.187·26-s + 0.823·28-s − 1.77·29-s + 0.269·31-s + 0.176·32-s − 0.426·34-s + 1.59·35-s + 1.12·37-s + 0.342·40-s + 0.0817·41-s − 1.37·43-s + 0.591·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6498 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6498 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.381327583\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.381327583\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 19 | \( 1 \) |
good | 5 | \( 1 - 2.16T + 5T^{2} \) |
| 7 | \( 1 - 4.36T + 7T^{2} \) |
| 11 | \( 1 - 3.92T + 11T^{2} \) |
| 13 | \( 1 + 0.954T + 13T^{2} \) |
| 17 | \( 1 + 2.48T + 17T^{2} \) |
| 23 | \( 1 - 7.07T + 23T^{2} \) |
| 29 | \( 1 + 9.55T + 29T^{2} \) |
| 31 | \( 1 - 1.50T + 31T^{2} \) |
| 37 | \( 1 - 6.81T + 37T^{2} \) |
| 41 | \( 1 - 0.523T + 41T^{2} \) |
| 43 | \( 1 + 9.04T + 43T^{2} \) |
| 47 | \( 1 - 4.35T + 47T^{2} \) |
| 53 | \( 1 + 8.02T + 53T^{2} \) |
| 59 | \( 1 - 13.8T + 59T^{2} \) |
| 61 | \( 1 - 5.14T + 61T^{2} \) |
| 67 | \( 1 - 6.24T + 67T^{2} \) |
| 71 | \( 1 - 2.47T + 71T^{2} \) |
| 73 | \( 1 + 3.95T + 73T^{2} \) |
| 79 | \( 1 + 13.7T + 79T^{2} \) |
| 83 | \( 1 - 6.89T + 83T^{2} \) |
| 89 | \( 1 + 5.84T + 89T^{2} \) |
| 97 | \( 1 + 15.9T + 97T^{2} \) |
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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
−7.958093324037008001350197059945, −7.10883686499281410843450720665, −6.59881964910597318776504948549, −5.64846302955453301298798343982, −5.23233192444687546124502300054, −4.45901117850776771263448677357, −3.82822337521080219649743455820, −2.62119691730666516079554003464, −1.84689608737822213192519642954, −1.23793985303189328356049940540,
1.23793985303189328356049940540, 1.84689608737822213192519642954, 2.62119691730666516079554003464, 3.82822337521080219649743455820, 4.45901117850776771263448677357, 5.23233192444687546124502300054, 5.64846302955453301298798343982, 6.59881964910597318776504948549, 7.10883686499281410843450720665, 7.958093324037008001350197059945