| L(s) = 1 | + 6.11·2-s + 30.6·3-s + 5.34·4-s − 25·5-s + 187.·6-s − 112.·7-s − 162.·8-s + 698.·9-s − 152.·10-s − 121·11-s + 163.·12-s + 90.1·13-s − 687.·14-s − 766.·15-s − 1.16e3·16-s + 205.·17-s + 4.26e3·18-s − 361·19-s − 133.·20-s − 3.45e3·21-s − 739.·22-s − 1.16e3·23-s − 4.99e3·24-s + 625·25-s + 551.·26-s + 1.39e4·27-s − 601.·28-s + ⋯ |
| L(s) = 1 | + 1.08·2-s + 1.96·3-s + 0.166·4-s − 0.447·5-s + 2.12·6-s − 0.867·7-s − 0.899·8-s + 2.87·9-s − 0.483·10-s − 0.301·11-s + 0.328·12-s + 0.148·13-s − 0.937·14-s − 0.880·15-s − 1.13·16-s + 0.172·17-s + 3.10·18-s − 0.229·19-s − 0.0746·20-s − 1.70·21-s − 0.325·22-s − 0.460·23-s − 1.77·24-s + 0.200·25-s + 0.159·26-s + 3.68·27-s − 0.144·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 25T \) |
| 11 | \( 1 + 121T \) |
| 19 | \( 1 + 361T \) |
| good | 2 | \( 1 - 6.11T + 32T^{2} \) |
| 3 | \( 1 - 30.6T + 243T^{2} \) |
| 7 | \( 1 + 112.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 90.1T + 3.71e5T^{2} \) |
| 17 | \( 1 - 205.T + 1.41e6T^{2} \) |
| 23 | \( 1 + 1.16e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.36e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.86e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.07e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.31e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.55e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.95e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.63e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.50e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.88e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.41e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.32e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.33e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.17e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.18e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.33e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.01e5T + 8.58e9T^{2} \) |
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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
−8.703149487410307738572903380883, −8.053392668626146564850703006225, −7.14874626839124930780778212239, −6.28550257607125368878718226316, −4.95731881868284753818987863429, −4.02481987987744372664582243683, −3.42841186516169968477931242262, −2.86825643661818821678809365661, −1.76124462343913403238499365301, 0,
1.76124462343913403238499365301, 2.86825643661818821678809365661, 3.42841186516169968477931242262, 4.02481987987744372664582243683, 4.95731881868284753818987863429, 6.28550257607125368878718226316, 7.14874626839124930780778212239, 8.053392668626146564850703006225, 8.703149487410307738572903380883
