| L(s) = 1 | − 7.82·2-s + 16.3·3-s + 29.2·4-s − 127.·6-s + 125.·7-s + 21.7·8-s + 22.8·9-s − 121·11-s + 476.·12-s − 532.·13-s − 981.·14-s − 1.10e3·16-s + 1.37e3·17-s − 178.·18-s − 554.·19-s + 2.04e3·21-s + 946.·22-s − 4.25e3·23-s + 353.·24-s + 4.16e3·26-s − 3.58e3·27-s + 3.66e3·28-s − 6.97e3·29-s + 3.13e3·31-s + 7.95e3·32-s − 1.97e3·33-s − 1.07e4·34-s + ⋯ |
| L(s) = 1 | − 1.38·2-s + 1.04·3-s + 0.913·4-s − 1.44·6-s + 0.967·7-s + 0.119·8-s + 0.0939·9-s − 0.301·11-s + 0.955·12-s − 0.873·13-s − 1.33·14-s − 1.07·16-s + 1.15·17-s − 0.129·18-s − 0.352·19-s + 1.01·21-s + 0.417·22-s − 1.67·23-s + 0.125·24-s + 1.20·26-s − 0.947·27-s + 0.883·28-s − 1.53·29-s + 0.584·31-s + 1.37·32-s − 0.315·33-s − 1.59·34-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 | \( 1 + 7.82T + 32T^{2} \) |
| 3 | \( 1 - 16.3T + 243T^{2} \) |
| 7 | \( 1 - 125.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 532.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.37e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 554.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 4.25e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 6.97e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.13e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.38e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 679.T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.72e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.51e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 9.54e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.75e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.05e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.87e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.25e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.37e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.86e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.22e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 8.15e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + 7.90e4T + 8.58e9T^{2} \) |
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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
−10.17039977011455523587304865210, −9.531386454569905765802208309325, −8.521259630283942249428802956931, −7.931457565361560382491676776073, −7.35331602170971723071813237297, −5.55894312072090814837395274771, −4.11486202829262416226946965468, −2.51270961467079109488587761555, −1.59451884625849978521285889452, 0,
1.59451884625849978521285889452, 2.51270961467079109488587761555, 4.11486202829262416226946965468, 5.55894312072090814837395274771, 7.35331602170971723071813237297, 7.931457565361560382491676776073, 8.521259630283942249428802956931, 9.531386454569905765802208309325, 10.17039977011455523587304865210
