L(s) = 1 | + 2-s − 0.0101·3-s + 4-s − 3.08·5-s − 0.0101·6-s + 7-s + 8-s − 2.99·9-s − 3.08·10-s − 6.48·11-s − 0.0101·12-s + 1.04·13-s + 14-s + 0.0312·15-s + 16-s + 2.51·17-s − 2.99·18-s − 3.08·20-s − 0.0101·21-s − 6.48·22-s − 8.22·23-s − 0.0101·24-s + 4.54·25-s + 1.04·26-s + 0.0606·27-s + 28-s + 5.21·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.00583·3-s + 0.5·4-s − 1.38·5-s − 0.00412·6-s + 0.377·7-s + 0.353·8-s − 0.999·9-s − 0.977·10-s − 1.95·11-s − 0.00291·12-s + 0.288·13-s + 0.267·14-s + 0.00805·15-s + 0.250·16-s + 0.609·17-s − 0.707·18-s − 0.690·20-s − 0.00220·21-s − 1.38·22-s − 1.71·23-s − 0.00206·24-s + 0.909·25-s + 0.203·26-s + 0.0116·27-s + 0.188·28-s + 0.969·29-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
\(L(1)\) |
\(\approx\) |
\(1.499322626\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.499322626\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.0101T + 3T^{2} \) |
| 5 | \( 1 + 3.08T + 5T^{2} \) |
| 11 | \( 1 + 6.48T + 11T^{2} \) |
| 13 | \( 1 - 1.04T + 13T^{2} \) |
| 17 | \( 1 - 2.51T + 17T^{2} \) |
| 23 | \( 1 + 8.22T + 23T^{2} \) |
| 29 | \( 1 - 5.21T + 29T^{2} \) |
| 31 | \( 1 - 8.22T + 31T^{2} \) |
| 37 | \( 1 - 3.78T + 37T^{2} \) |
| 41 | \( 1 + 6.35T + 41T^{2} \) |
| 43 | \( 1 - 1.33T + 43T^{2} \) |
| 47 | \( 1 + 2.96T + 47T^{2} \) |
| 53 | \( 1 + 0.538T + 53T^{2} \) |
| 59 | \( 1 - 11.1T + 59T^{2} \) |
| 61 | \( 1 + 8.50T + 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 - 4.19T + 71T^{2} \) |
| 73 | \( 1 - 10.6T + 73T^{2} \) |
| 79 | \( 1 - 10.7T + 79T^{2} \) |
| 83 | \( 1 + 6.72T + 83T^{2} \) |
| 89 | \( 1 - 2.05T + 89T^{2} \) |
| 97 | \( 1 + 18.9T + 97T^{2} \) |
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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
−8.162037435677346355826444997200, −7.79423318784676739352027319813, −6.75896386476017978640506460140, −5.89654493459658525548749210230, −5.21307608281337588639249773405, −4.58425197738049533089469635854, −3.72525956957374971705617896150, −2.98413417154617245154970683052, −2.27747185731617362872611707513, −0.56740017488298460559182091942,
0.56740017488298460559182091942, 2.27747185731617362872611707513, 2.98413417154617245154970683052, 3.72525956957374971705617896150, 4.58425197738049533089469635854, 5.21307608281337588639249773405, 5.89654493459658525548749210230, 6.75896386476017978640506460140, 7.79423318784676739352027319813, 8.162037435677346355826444997200