| L(s) = 1 | + 2·3-s − 2·5-s + 9-s + 4·11-s + 2·13-s − 4·15-s + 2·17-s − 25-s − 4·27-s − 8·29-s − 8·31-s + 8·33-s − 4·37-s + 4·39-s + 10·41-s + 2·43-s − 2·45-s + 4·51-s + 8·53-s − 8·55-s + 14·59-s − 4·65-s − 8·67-s + 8·71-s − 6·73-s − 2·75-s − 79-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s + 1/3·9-s + 1.20·11-s + 0.554·13-s − 1.03·15-s + 0.485·17-s − 1/5·25-s − 0.769·27-s − 1.48·29-s − 1.43·31-s + 1.39·33-s − 0.657·37-s + 0.640·39-s + 1.56·41-s + 0.304·43-s − 0.298·45-s + 0.560·51-s + 1.09·53-s − 1.07·55-s + 1.82·59-s − 0.496·65-s − 0.977·67-s + 0.949·71-s − 0.702·73-s − 0.230·75-s − 0.112·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 247744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 247744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.975428539\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.975428539\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 79 | \( 1 + T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−12.94297565909099, −12.40734074880842, −11.85026963652845, −11.59655806108939, −10.92817079300391, −10.79371663640410, −9.794168649893828, −9.523527931868773, −9.055137704799707, −8.641338551256807, −8.285994491284539, −7.625718790857969, −7.374300430563084, −6.949757900176598, −6.169213804183642, −5.678151875336968, −5.250387359589544, −4.204498203775853, −3.957626526411805, −3.653546629337547, −3.166672869360208, −2.399750664872476, −1.870193350536109, −1.249480242674335, −0.4425460187563591,
0.4425460187563591, 1.249480242674335, 1.870193350536109, 2.399750664872476, 3.166672869360208, 3.653546629337547, 3.957626526411805, 4.204498203775853, 5.250387359589544, 5.678151875336968, 6.169213804183642, 6.949757900176598, 7.374300430563084, 7.625718790857969, 8.285994491284539, 8.641338551256807, 9.055137704799707, 9.523527931868773, 9.794168649893828, 10.79371663640410, 10.92817079300391, 11.59655806108939, 11.85026963652845, 12.40734074880842, 12.94297565909099
