L(s) = 1 | − 2-s + 3.32·3-s + 4-s + 5-s − 3.32·6-s − 2.87·7-s − 8-s + 8.06·9-s − 10-s + 3.79·11-s + 3.32·12-s − 2.47·13-s + 2.87·14-s + 3.32·15-s + 16-s + 2.94·17-s − 8.06·18-s + 0.783·19-s + 20-s − 9.57·21-s − 3.79·22-s − 0.0265·23-s − 3.32·24-s + 25-s + 2.47·26-s + 16.8·27-s − 2.87·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.92·3-s + 0.5·4-s + 0.447·5-s − 1.35·6-s − 1.08·7-s − 0.353·8-s + 2.68·9-s − 0.316·10-s + 1.14·11-s + 0.960·12-s − 0.687·13-s + 0.769·14-s + 0.859·15-s + 0.250·16-s + 0.713·17-s − 1.90·18-s + 0.179·19-s + 0.223·20-s − 2.08·21-s − 0.809·22-s − 0.00552·23-s − 0.679·24-s + 0.200·25-s + 0.485·26-s + 3.24·27-s − 0.543·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.129015877\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.129015877\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 + T \) |
good | 3 | \( 1 - 3.32T + 3T^{2} \) |
| 7 | \( 1 + 2.87T + 7T^{2} \) |
| 11 | \( 1 - 3.79T + 11T^{2} \) |
| 13 | \( 1 + 2.47T + 13T^{2} \) |
| 17 | \( 1 - 2.94T + 17T^{2} \) |
| 19 | \( 1 - 0.783T + 19T^{2} \) |
| 23 | \( 1 + 0.0265T + 23T^{2} \) |
| 29 | \( 1 + 8.26T + 29T^{2} \) |
| 31 | \( 1 - 0.619T + 31T^{2} \) |
| 37 | \( 1 - 1.77T + 37T^{2} \) |
| 41 | \( 1 - 9.24T + 41T^{2} \) |
| 43 | \( 1 - 1.77T + 43T^{2} \) |
| 47 | \( 1 - 10.6T + 47T^{2} \) |
| 53 | \( 1 - 5.35T + 53T^{2} \) |
| 59 | \( 1 - 4.82T + 59T^{2} \) |
| 61 | \( 1 - 0.650T + 61T^{2} \) |
| 67 | \( 1 + 10.3T + 67T^{2} \) |
| 71 | \( 1 - 1.13T + 71T^{2} \) |
| 73 | \( 1 + 3.11T + 73T^{2} \) |
| 79 | \( 1 - 5.43T + 79T^{2} \) |
| 83 | \( 1 + 7.44T + 83T^{2} \) |
| 89 | \( 1 - 4.32T + 89T^{2} \) |
| 97 | \( 1 + 6.14T + 97T^{2} \) |
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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
−8.779675367023502718135561044031, −7.64099326812139232245187955127, −7.35639611603755552375279382706, −6.57071611158746708661949439721, −5.69475691390137872378196149978, −4.23502999377290986469651498710, −3.58326288404238060802716210931, −2.81240389417898116718266533902, −2.11222689304476523255846115740, −1.08853994531427669139110634633,
1.08853994531427669139110634633, 2.11222689304476523255846115740, 2.81240389417898116718266533902, 3.58326288404238060802716210931, 4.23502999377290986469651498710, 5.69475691390137872378196149978, 6.57071611158746708661949439721, 7.35639611603755552375279382706, 7.64099326812139232245187955127, 8.779675367023502718135561044031