| L(s) = 1 | − 1.45·2-s + 0.114·4-s − 2.59·5-s − 2.00·7-s + 2.74·8-s + 3.77·10-s + 11-s − 4.45·13-s + 2.92·14-s − 4.21·16-s − 4.54·17-s + 19-s − 0.297·20-s − 1.45·22-s − 7.48·23-s + 1.72·25-s + 6.48·26-s − 0.230·28-s + 3.17·29-s + 9.34·31-s + 0.647·32-s + 6.60·34-s + 5.21·35-s − 6.84·37-s − 1.45·38-s − 7.10·40-s − 0.644·41-s + ⋯ |
| L(s) = 1 | − 1.02·2-s + 0.0572·4-s − 1.15·5-s − 0.759·7-s + 0.969·8-s + 1.19·10-s + 0.301·11-s − 1.23·13-s + 0.780·14-s − 1.05·16-s − 1.10·17-s + 0.229·19-s − 0.0664·20-s − 0.310·22-s − 1.56·23-s + 0.344·25-s + 1.27·26-s − 0.0435·28-s + 0.589·29-s + 1.67·31-s + 0.114·32-s + 1.13·34-s + 0.880·35-s − 1.12·37-s − 0.235·38-s − 1.12·40-s − 0.100·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1881 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1881 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2476988596\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2476988596\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 1.45T + 2T^{2} \) |
| 5 | \( 1 + 2.59T + 5T^{2} \) |
| 7 | \( 1 + 2.00T + 7T^{2} \) |
| 13 | \( 1 + 4.45T + 13T^{2} \) |
| 17 | \( 1 + 4.54T + 17T^{2} \) |
| 23 | \( 1 + 7.48T + 23T^{2} \) |
| 29 | \( 1 - 3.17T + 29T^{2} \) |
| 31 | \( 1 - 9.34T + 31T^{2} \) |
| 37 | \( 1 + 6.84T + 37T^{2} \) |
| 41 | \( 1 + 0.644T + 41T^{2} \) |
| 43 | \( 1 + 8.07T + 43T^{2} \) |
| 47 | \( 1 + 11.6T + 47T^{2} \) |
| 53 | \( 1 - 5.53T + 53T^{2} \) |
| 59 | \( 1 + 9.16T + 59T^{2} \) |
| 61 | \( 1 - 9.45T + 61T^{2} \) |
| 67 | \( 1 - 0.113T + 67T^{2} \) |
| 71 | \( 1 - 9.84T + 71T^{2} \) |
| 73 | \( 1 + 2.38T + 73T^{2} \) |
| 79 | \( 1 + 2.01T + 79T^{2} \) |
| 83 | \( 1 - 2.90T + 83T^{2} \) |
| 89 | \( 1 - 8.82T + 89T^{2} \) |
| 97 | \( 1 - 11.0T + 97T^{2} \) |
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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
−9.236354673463371500147588553314, −8.307674940757123310161751973296, −7.973147994341432246150453209501, −7.00387542715732833414903504755, −6.46741124524642745373776241764, −4.93098622258183975261681190345, −4.29953245786613256529485147732, −3.32987825402886191946760069002, −2.03200904359066458909134711951, −0.37616945552612121761066263649,
0.37616945552612121761066263649, 2.03200904359066458909134711951, 3.32987825402886191946760069002, 4.29953245786613256529485147732, 4.93098622258183975261681190345, 6.46741124524642745373776241764, 7.00387542715732833414903504755, 7.973147994341432246150453209501, 8.307674940757123310161751973296, 9.236354673463371500147588553314
