| L(s) = 1 | + 3·3-s + 4·5-s − 7-s + 6·9-s − 2·11-s − 5·13-s + 12·15-s + 4·19-s − 3·21-s + 23-s + 11·25-s + 9·27-s + 3·29-s + 5·31-s − 6·33-s − 4·35-s − 4·37-s − 15·39-s + 5·41-s + 4·43-s + 24·45-s − 11·47-s + 49-s − 8·55-s + 12·57-s + 12·59-s + 6·61-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 1.78·5-s − 0.377·7-s + 2·9-s − 0.603·11-s − 1.38·13-s + 3.09·15-s + 0.917·19-s − 0.654·21-s + 0.208·23-s + 11/5·25-s + 1.73·27-s + 0.557·29-s + 0.898·31-s − 1.04·33-s − 0.676·35-s − 0.657·37-s − 2.40·39-s + 0.780·41-s + 0.609·43-s + 3.57·45-s − 1.60·47-s + 1/7·49-s − 1.07·55-s + 1.58·57-s + 1.56·59-s + 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.804408370\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.804408370\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 11 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−16.52296840526787, −15.93600517039981, −15.29071505978200, −14.68130768491984, −14.12815749303628, −13.92083928043469, −13.30353183341918, −12.77533686694304, −12.38662414996057, −11.26804200732754, −10.14561901439347, −10.07960432108480, −9.482018679074787, −9.143195682640308, −8.264940795792882, −7.806501113722858, −6.885030143357390, −6.578849385072158, −5.316099501555570, −5.116812538412009, −3.986007163726490, −3.007044013522224, −2.548706019070745, −2.115647349838402, −1.077427571647196,
1.077427571647196, 2.115647349838402, 2.548706019070745, 3.007044013522224, 3.986007163726490, 5.116812538412009, 5.316099501555570, 6.578849385072158, 6.885030143357390, 7.806501113722858, 8.264940795792882, 9.143195682640308, 9.482018679074787, 10.07960432108480, 10.14561901439347, 11.26804200732754, 12.38662414996057, 12.77533686694304, 13.30353183341918, 13.92083928043469, 14.12815749303628, 14.68130768491984, 15.29071505978200, 15.93600517039981, 16.52296840526787
