| L(s) = 1 | − 6.40·2-s + 15.0·3-s + 9.01·4-s − 96.4·6-s + 122.·7-s + 147.·8-s − 16.0·9-s + 121·11-s + 135.·12-s + 1.04e3·13-s − 783.·14-s − 1.23e3·16-s − 400.·17-s + 102.·18-s + 581.·19-s + 1.84e3·21-s − 774.·22-s − 66.9·23-s + 2.21e3·24-s − 6.67e3·26-s − 3.90e3·27-s + 1.10e3·28-s + 6.78e3·29-s − 3.86e3·31-s + 3.17e3·32-s + 1.82e3·33-s + 2.56e3·34-s + ⋯ |
| L(s) = 1 | − 1.13·2-s + 0.966·3-s + 0.281·4-s − 1.09·6-s + 0.944·7-s + 0.813·8-s − 0.0660·9-s + 0.301·11-s + 0.272·12-s + 1.71·13-s − 1.06·14-s − 1.20·16-s − 0.336·17-s + 0.0747·18-s + 0.369·19-s + 0.912·21-s − 0.341·22-s − 0.0263·23-s + 0.785·24-s − 1.93·26-s − 1.03·27-s + 0.266·28-s + 1.49·29-s − 0.721·31-s + 0.548·32-s + 0.291·33-s + 0.380·34-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
| \(L(3)\) |
\(\approx\) |
\(1.819386051\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.819386051\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 6.40T + 32T^{2} \) |
| 3 | \( 1 - 15.0T + 243T^{2} \) |
| 7 | \( 1 - 122.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 1.04e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 400.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 581.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 66.9T + 6.43e6T^{2} \) |
| 29 | \( 1 - 6.78e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.86e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.45e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 5.66e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.85e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.73e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.68e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.98e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.46e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.99e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.49e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 8.17e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.68e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.00e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.05e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.38e5T + 8.58e9T^{2} \) |
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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
−10.95855069317234376046220562417, −9.797582486940432675085500855800, −8.875019838031556005467476665867, −8.358185214759329605837220393391, −7.73108304073187284205135481735, −6.33285056544562519267466022085, −4.74567367097077819783415875361, −3.50486869418495695787600909551, −1.96552347096598980649556316624, −0.941195341659993384756296357691,
0.941195341659993384756296357691, 1.96552347096598980649556316624, 3.50486869418495695787600909551, 4.74567367097077819783415875361, 6.33285056544562519267466022085, 7.73108304073187284205135481735, 8.358185214759329605837220393391, 8.875019838031556005467476665867, 9.797582486940432675085500855800, 10.95855069317234376046220562417
