| L(s) = 1 | − 2-s − 15·4-s − 49·7-s + 31·8-s + 81·9-s − 206·11-s + 49·14-s + 209·16-s − 81·18-s + 206·22-s + 734·23-s + 735·28-s + 1.23e3·29-s − 705·32-s − 1.21e3·36-s + 1.29e3·37-s + 334·43-s + 3.09e3·44-s − 734·46-s + 2.40e3·49-s + 5.58e3·53-s − 1.51e3·56-s − 1.23e3·58-s − 3.96e3·63-s − 2.63e3·64-s − 4.94e3·67-s + 2.91e3·71-s + ⋯ |
| L(s) = 1 | − 1/4·2-s − 0.937·4-s − 7-s + 0.484·8-s + 9-s − 1.70·11-s + 1/4·14-s + 0.816·16-s − 1/4·18-s + 0.425·22-s + 1.38·23-s + 0.937·28-s + 1.46·29-s − 0.688·32-s − 0.937·36-s + 0.945·37-s + 0.180·43-s + 1.59·44-s − 0.346·46-s + 49-s + 1.98·53-s − 0.484·56-s − 0.366·58-s − 63-s − 0.644·64-s − 1.10·67-s + 0.578·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.9867361074\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9867361074\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 + p^{2} T \) |
| good | 2 | \( 1 + T + p^{4} T^{2} \) |
| 3 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 11 | \( 1 + 206 T + p^{4} T^{2} \) |
| 13 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 17 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 19 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 23 | \( 1 - 734 T + p^{4} T^{2} \) |
| 29 | \( 1 - 1234 T + p^{4} T^{2} \) |
| 31 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 37 | \( 1 - 1294 T + p^{4} T^{2} \) |
| 41 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 43 | \( 1 - 334 T + p^{4} T^{2} \) |
| 47 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 53 | \( 1 - 5582 T + p^{4} T^{2} \) |
| 59 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 61 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 67 | \( 1 + 4946 T + p^{4} T^{2} \) |
| 71 | \( 1 - 2914 T + p^{4} T^{2} \) |
| 73 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 79 | \( 1 + 3646 T + p^{4} T^{2} \) |
| 83 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 89 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 97 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
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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
−12.37396704661604279665517269154, −10.60038750242078630535181495454, −10.05956102446464702736683688807, −9.107635617850217560071374193435, −7.980395253428300847907467820821, −6.94646205341092811241512731591, −5.43088154739179142616086833158, −4.35669123879986398134206051716, −2.88269957781567645021396316359, −0.72213723308991920344142610595,
0.72213723308991920344142610595, 2.88269957781567645021396316359, 4.35669123879986398134206051716, 5.43088154739179142616086833158, 6.94646205341092811241512731591, 7.980395253428300847907467820821, 9.107635617850217560071374193435, 10.05956102446464702736683688807, 10.60038750242078630535181495454, 12.37396704661604279665517269154
