| L(s) = 1 | + 8.50·2-s + 13.0·3-s + 40.3·4-s + 110.·6-s − 217.·7-s + 70.7·8-s − 73.1·9-s − 121·11-s + 525.·12-s − 747.·13-s − 1.84e3·14-s − 688.·16-s + 677.·17-s − 622.·18-s − 1.91e3·19-s − 2.83e3·21-s − 1.02e3·22-s + 3.51e3·23-s + 921.·24-s − 6.35e3·26-s − 4.12e3·27-s − 8.76e3·28-s + 4.65e3·29-s + 371.·31-s − 8.11e3·32-s − 1.57e3·33-s + 5.75e3·34-s + ⋯ |
| L(s) = 1 | + 1.50·2-s + 0.835·3-s + 1.25·4-s + 1.25·6-s − 1.67·7-s + 0.390·8-s − 0.301·9-s − 0.301·11-s + 1.05·12-s − 1.22·13-s − 2.52·14-s − 0.672·16-s + 0.568·17-s − 0.452·18-s − 1.21·19-s − 1.40·21-s − 0.453·22-s + 1.38·23-s + 0.326·24-s − 1.84·26-s − 1.08·27-s − 2.11·28-s + 1.02·29-s + 0.0694·31-s − 1.40·32-s − 0.252·33-s + 0.854·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 + 121T \) |
| good | 2 | \( 1 - 8.50T + 32T^{2} \) |
| 3 | \( 1 - 13.0T + 243T^{2} \) |
| 7 | \( 1 + 217.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 747.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 677.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.91e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.51e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.65e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 371.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.72e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.63e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.92e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 5.24e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.96e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.30e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.71e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.42e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.37e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.19e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.58e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 8.84e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.03e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.35e4T + 8.58e9T^{2} \) |
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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
−10.64016212738035761929538159773, −9.584016975291023843310034646917, −8.729456011861780398706411027774, −7.26904326523478756765653438858, −6.39355984167046833416505536917, −5.37276240291423212759507831561, −4.13883833176079866951136693815, −2.99627624557586362294187700719, −2.62008264986407045343776696543, 0,
2.62008264986407045343776696543, 2.99627624557586362294187700719, 4.13883833176079866951136693815, 5.37276240291423212759507831561, 6.39355984167046833416505536917, 7.26904326523478756765653438858, 8.729456011861780398706411027774, 9.584016975291023843310034646917, 10.64016212738035761929538159773
