| L(s) = 1 | + 3·3-s + 7.29·5-s − 5.87·7-s + 9·9-s − 51.1·11-s + 13·13-s + 21.8·15-s − 73.7·17-s + 59.9·19-s − 17.6·21-s − 69.8·23-s − 71.8·25-s + 27·27-s + 294.·29-s + 334.·31-s − 153.·33-s − 42.8·35-s − 261.·37-s + 39·39-s + 222.·41-s + 79.2·43-s + 65.6·45-s + 584.·47-s − 308.·49-s − 221.·51-s − 465.·53-s − 373.·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.652·5-s − 0.317·7-s + 0.333·9-s − 1.40·11-s + 0.277·13-s + 0.376·15-s − 1.05·17-s + 0.723·19-s − 0.183·21-s − 0.633·23-s − 0.574·25-s + 0.192·27-s + 1.88·29-s + 1.93·31-s − 0.809·33-s − 0.206·35-s − 1.16·37-s + 0.160·39-s + 0.848·41-s + 0.281·43-s + 0.217·45-s + 1.81·47-s − 0.899·49-s − 0.607·51-s − 1.20·53-s − 0.914·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 13 | \( 1 - 13T \) |
| good | 5 | \( 1 - 7.29T + 125T^{2} \) |
| 7 | \( 1 + 5.87T + 343T^{2} \) |
| 11 | \( 1 + 51.1T + 1.33e3T^{2} \) |
| 17 | \( 1 + 73.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 59.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 69.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 294.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 334.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 261.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 222.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 79.2T + 7.95e4T^{2} \) |
| 47 | \( 1 - 584.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 465.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 530.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 548.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 384.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 307.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 844.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 30.1T + 4.93e5T^{2} \) |
| 83 | \( 1 - 19.5T + 5.71e5T^{2} \) |
| 89 | \( 1 + 513.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 787.T + 9.12e5T^{2} \) |
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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
−8.179473996815969001114877967456, −7.60078431314087134845504759516, −6.56193804070687605551012393110, −5.96806141901658871378813794929, −4.97285651560781531315492262194, −4.23280792649944375158581073829, −2.92571494687994683160409414772, −2.53332512607841829737139523671, −1.35492040380771831997716735979, 0,
1.35492040380771831997716735979, 2.53332512607841829737139523671, 2.92571494687994683160409414772, 4.23280792649944375158581073829, 4.97285651560781531315492262194, 5.96806141901658871378813794929, 6.56193804070687605551012393110, 7.60078431314087134845504759516, 8.179473996815969001114877967456
