| L(s) = 1 | − 2-s − 2.70·3-s + 4-s + 5-s + 2.70·6-s − 1.21·7-s − 8-s + 4.29·9-s − 10-s − 5.34·11-s − 2.70·12-s + 4.99·13-s + 1.21·14-s − 2.70·15-s + 16-s + 1.52·17-s − 4.29·18-s − 5.54·19-s + 20-s + 3.27·21-s + 5.34·22-s − 2.17·23-s + 2.70·24-s + 25-s − 4.99·26-s − 3.49·27-s − 1.21·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.55·3-s + 0.5·4-s + 0.447·5-s + 1.10·6-s − 0.457·7-s − 0.353·8-s + 1.43·9-s − 0.316·10-s − 1.61·11-s − 0.779·12-s + 1.38·13-s + 0.323·14-s − 0.697·15-s + 0.250·16-s + 0.369·17-s − 1.01·18-s − 1.27·19-s + 0.223·20-s + 0.714·21-s + 1.13·22-s − 0.453·23-s + 0.551·24-s + 0.200·25-s − 0.979·26-s − 0.672·27-s − 0.228·28-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.70T + 3T^{2} \) |
| 7 | \( 1 + 1.21T + 7T^{2} \) |
| 11 | \( 1 + 5.34T + 11T^{2} \) |
| 13 | \( 1 - 4.99T + 13T^{2} \) |
| 17 | \( 1 - 1.52T + 17T^{2} \) |
| 19 | \( 1 + 5.54T + 19T^{2} \) |
| 23 | \( 1 + 2.17T + 23T^{2} \) |
| 29 | \( 1 - 5.67T + 29T^{2} \) |
| 31 | \( 1 - 7.09T + 31T^{2} \) |
| 37 | \( 1 - 1.20T + 37T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 + 12.2T + 43T^{2} \) |
| 47 | \( 1 - 12.4T + 47T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 - 3.79T + 59T^{2} \) |
| 61 | \( 1 + 6.62T + 61T^{2} \) |
| 67 | \( 1 + 7.49T + 67T^{2} \) |
| 71 | \( 1 + 0.842T + 71T^{2} \) |
| 73 | \( 1 - 8.60T + 73T^{2} \) |
| 79 | \( 1 - 4.99T + 79T^{2} \) |
| 83 | \( 1 + 1.39T + 83T^{2} \) |
| 89 | \( 1 - 1.31T + 89T^{2} \) |
| 97 | \( 1 + 9.84T + 97T^{2} \) |
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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
−8.306076407224326578088793895469, −7.18131805348426792572176132301, −6.38279993580015570072060455852, −6.09086201336506549940768300490, −5.30929848578303671197409295493, −4.55321360679107665206907335551, −3.30294112490003428403205993015, −2.21741838233280892897559126398, −1.02625507922840265802455436450, 0,
1.02625507922840265802455436450, 2.21741838233280892897559126398, 3.30294112490003428403205993015, 4.55321360679107665206907335551, 5.30929848578303671197409295493, 6.09086201336506549940768300490, 6.38279993580015570072060455852, 7.18131805348426792572176132301, 8.306076407224326578088793895469
