| L(s) = 1 | + 2-s − 2.67·3-s + 4-s − 5-s − 2.67·6-s − 7-s + 8-s + 4.15·9-s − 10-s − 2.67·12-s + 3.11·13-s − 14-s + 2.67·15-s + 16-s + 0.564·17-s + 4.15·18-s − 7.52·19-s − 20-s + 2.67·21-s + 4.63·23-s − 2.67·24-s + 25-s + 3.11·26-s − 3.09·27-s − 28-s − 6.24·29-s + 2.67·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.54·3-s + 0.5·4-s − 0.447·5-s − 1.09·6-s − 0.377·7-s + 0.353·8-s + 1.38·9-s − 0.316·10-s − 0.772·12-s + 0.864·13-s − 0.267·14-s + 0.690·15-s + 0.250·16-s + 0.136·17-s + 0.980·18-s − 1.72·19-s − 0.223·20-s + 0.583·21-s + 0.966·23-s − 0.546·24-s + 0.200·25-s + 0.611·26-s − 0.596·27-s − 0.188·28-s − 1.15·29-s + 0.488·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.275216196\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.275216196\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 2.67T + 3T^{2} \) |
| 13 | \( 1 - 3.11T + 13T^{2} \) |
| 17 | \( 1 - 0.564T + 17T^{2} \) |
| 19 | \( 1 + 7.52T + 19T^{2} \) |
| 23 | \( 1 - 4.63T + 23T^{2} \) |
| 29 | \( 1 + 6.24T + 29T^{2} \) |
| 31 | \( 1 - 2.72T + 31T^{2} \) |
| 37 | \( 1 + 0.240T + 37T^{2} \) |
| 41 | \( 1 - 2.28T + 41T^{2} \) |
| 43 | \( 1 - 3.78T + 43T^{2} \) |
| 47 | \( 1 + 0.981T + 47T^{2} \) |
| 53 | \( 1 + 1.14T + 53T^{2} \) |
| 59 | \( 1 - 8.79T + 59T^{2} \) |
| 61 | \( 1 + 11.9T + 61T^{2} \) |
| 67 | \( 1 + 6.80T + 67T^{2} \) |
| 71 | \( 1 - 11.4T + 71T^{2} \) |
| 73 | \( 1 + 13.2T + 73T^{2} \) |
| 79 | \( 1 - 15.3T + 79T^{2} \) |
| 83 | \( 1 + 10.5T + 83T^{2} \) |
| 89 | \( 1 + 4.33T + 89T^{2} \) |
| 97 | \( 1 + 9.05T + 97T^{2} \) |
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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
−7.46062929677355646876561250198, −6.81517478865952322084505093756, −6.23658945777095304082561145657, −5.81094893843958128397835831951, −5.03969160221207235576394661061, −4.34505833478909161130487388078, −3.79591554086456601514667164294, −2.81038408745714698031185207386, −1.61997039094973048434086533981, −0.54623396562703542390174169692,
0.54623396562703542390174169692, 1.61997039094973048434086533981, 2.81038408745714698031185207386, 3.79591554086456601514667164294, 4.34505833478909161130487388078, 5.03969160221207235576394661061, 5.81094893843958128397835831951, 6.23658945777095304082561145657, 6.81517478865952322084505093756, 7.46062929677355646876561250198
