| L(s) = 1 | − 2-s − 0.572·3-s + 4-s + 2.21·5-s + 0.572·6-s + 7-s − 8-s − 2.67·9-s − 2.21·10-s + 6.09·11-s − 0.572·12-s − 6.15·13-s − 14-s − 1.26·15-s + 16-s − 1.73·17-s + 2.67·18-s + 2.21·20-s − 0.572·21-s − 6.09·22-s − 3.38·23-s + 0.572·24-s − 0.0738·25-s + 6.15·26-s + 3.24·27-s + 28-s + 2.18·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.330·3-s + 0.5·4-s + 0.992·5-s + 0.233·6-s + 0.377·7-s − 0.353·8-s − 0.890·9-s − 0.701·10-s + 1.83·11-s − 0.165·12-s − 1.70·13-s − 0.267·14-s − 0.327·15-s + 0.250·16-s − 0.421·17-s + 0.629·18-s + 0.496·20-s − 0.124·21-s − 1.29·22-s − 0.706·23-s + 0.116·24-s − 0.0147·25-s + 1.20·26-s + 0.624·27-s + 0.188·28-s + 0.406·29-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
| \(L(1)\) |
\(\approx\) |
\(1.364626522\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.364626522\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.572T + 3T^{2} \) |
| 5 | \( 1 - 2.21T + 5T^{2} \) |
| 11 | \( 1 - 6.09T + 11T^{2} \) |
| 13 | \( 1 + 6.15T + 13T^{2} \) |
| 17 | \( 1 + 1.73T + 17T^{2} \) |
| 23 | \( 1 + 3.38T + 23T^{2} \) |
| 29 | \( 1 - 2.18T + 29T^{2} \) |
| 31 | \( 1 - 0.977T + 31T^{2} \) |
| 37 | \( 1 + 11.9T + 37T^{2} \) |
| 41 | \( 1 - 8.25T + 41T^{2} \) |
| 43 | \( 1 - 3.28T + 43T^{2} \) |
| 47 | \( 1 - 9.95T + 47T^{2} \) |
| 53 | \( 1 + 3.81T + 53T^{2} \) |
| 59 | \( 1 - 3.92T + 59T^{2} \) |
| 61 | \( 1 - 1.40T + 61T^{2} \) |
| 67 | \( 1 - 1.83T + 67T^{2} \) |
| 71 | \( 1 - 16.2T + 71T^{2} \) |
| 73 | \( 1 + 8.95T + 73T^{2} \) |
| 79 | \( 1 - 15.3T + 79T^{2} \) |
| 83 | \( 1 - 11.7T + 83T^{2} \) |
| 89 | \( 1 - 13.2T + 89T^{2} \) |
| 97 | \( 1 + 11.4T + 97T^{2} \) |
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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
−8.378019603391180417853928410780, −7.48753691061723518768545606663, −6.74860998977473184086721079431, −6.18679807292580723049552737584, −5.49199966533871998610029804681, −4.69856080264032486654061139091, −3.66912554253286522136094596962, −2.42981467811465690680572814730, −1.92820745077340497621570015599, −0.71207860847016851933863103688,
0.71207860847016851933863103688, 1.92820745077340497621570015599, 2.42981467811465690680572814730, 3.66912554253286522136094596962, 4.69856080264032486654061139091, 5.49199966533871998610029804681, 6.18679807292580723049552737584, 6.74860998977473184086721079431, 7.48753691061723518768545606663, 8.378019603391180417853928410780
