| L(s) = 1 | − 2-s + 2.34·3-s + 4-s − 5-s − 2.34·6-s − 2.08·7-s − 8-s + 2.49·9-s + 10-s + 6.04·11-s + 2.34·12-s + 1.31·13-s + 2.08·14-s − 2.34·15-s + 16-s − 5.09·17-s − 2.49·18-s − 5.37·19-s − 20-s − 4.88·21-s − 6.04·22-s − 3.94·23-s − 2.34·24-s + 25-s − 1.31·26-s − 1.17·27-s − 2.08·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.35·3-s + 0.5·4-s − 0.447·5-s − 0.957·6-s − 0.787·7-s − 0.353·8-s + 0.833·9-s + 0.316·10-s + 1.82·11-s + 0.676·12-s + 0.365·13-s + 0.556·14-s − 0.605·15-s + 0.250·16-s − 1.23·17-s − 0.589·18-s − 1.23·19-s − 0.223·20-s − 1.06·21-s − 1.28·22-s − 0.823·23-s − 0.478·24-s + 0.200·25-s − 0.258·26-s − 0.226·27-s − 0.393·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 601 | \( 1 + T \) |
| good | 3 | \( 1 - 2.34T + 3T^{2} \) |
| 7 | \( 1 + 2.08T + 7T^{2} \) |
| 11 | \( 1 - 6.04T + 11T^{2} \) |
| 13 | \( 1 - 1.31T + 13T^{2} \) |
| 17 | \( 1 + 5.09T + 17T^{2} \) |
| 19 | \( 1 + 5.37T + 19T^{2} \) |
| 23 | \( 1 + 3.94T + 23T^{2} \) |
| 29 | \( 1 + 3.93T + 29T^{2} \) |
| 31 | \( 1 - 5.38T + 31T^{2} \) |
| 37 | \( 1 + 0.767T + 37T^{2} \) |
| 41 | \( 1 + 8.66T + 41T^{2} \) |
| 43 | \( 1 - 10.7T + 43T^{2} \) |
| 47 | \( 1 - 5.34T + 47T^{2} \) |
| 53 | \( 1 + 8.02T + 53T^{2} \) |
| 59 | \( 1 - 4.16T + 59T^{2} \) |
| 61 | \( 1 + 0.0265T + 61T^{2} \) |
| 67 | \( 1 + 10.1T + 67T^{2} \) |
| 71 | \( 1 - 11.0T + 71T^{2} \) |
| 73 | \( 1 + 15.6T + 73T^{2} \) |
| 79 | \( 1 + 3.33T + 79T^{2} \) |
| 83 | \( 1 + 5.78T + 83T^{2} \) |
| 89 | \( 1 + 6.35T + 89T^{2} \) |
| 97 | \( 1 + 7.21T + 97T^{2} \) |
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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
−7.939635006754268467910664963627, −7.10384940850042979685851660210, −6.52604679379431380579947380746, −6.00962218980009071980828017824, −4.29828974954643514963480103606, −3.96378944039239164621445171734, −3.15142166719151338434693057673, −2.28865156727274471964746832808, −1.47643626042951329661149028198, 0,
1.47643626042951329661149028198, 2.28865156727274471964746832808, 3.15142166719151338434693057673, 3.96378944039239164621445171734, 4.29828974954643514963480103606, 6.00962218980009071980828017824, 6.52604679379431380579947380746, 7.10384940850042979685851660210, 7.939635006754268467910664963627
