| L(s) = 1 | − 3·3-s − 20.1·5-s + 19.6·7-s + 9·9-s − 13.5·11-s + 13·13-s + 60.4·15-s + 116.·17-s − 68.1·19-s − 59.0·21-s − 122.·23-s + 280.·25-s − 27·27-s + 204.·29-s + 194.·31-s + 40.6·33-s − 396.·35-s − 142.·37-s − 39·39-s − 175.·41-s + 219.·43-s − 181.·45-s − 236.·47-s + 45.0·49-s − 349.·51-s − 628.·53-s + 273.·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.80·5-s + 1.06·7-s + 0.333·9-s − 0.371·11-s + 0.277·13-s + 1.04·15-s + 1.66·17-s − 0.822·19-s − 0.614·21-s − 1.10·23-s + 2.24·25-s − 0.192·27-s + 1.30·29-s + 1.12·31-s + 0.214·33-s − 1.91·35-s − 0.634·37-s − 0.160·39-s − 0.666·41-s + 0.778·43-s − 0.600·45-s − 0.733·47-s + 0.131·49-s − 0.958·51-s − 1.62·53-s + 0.669·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 20.1T + 125T^{2} \) |
| 7 | \( 1 - 19.6T + 343T^{2} \) |
| 11 | \( 1 + 13.5T + 1.33e3T^{2} \) |
| 17 | \( 1 - 116.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 68.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 122.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 204.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 194.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 142.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 175.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 219.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 236.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 628.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 446.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 224.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 165.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 902.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 15.1T + 3.89e5T^{2} \) |
| 79 | \( 1 + 670.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.04e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 562.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.64e3T + 9.12e5T^{2} \) |
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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
−10.08943501116698990748926447815, −8.421177813158901596650689205879, −8.079800066106881357114366719677, −7.32444335473090235270531615879, −6.13633467934958653813781163805, −4.89882587421777491316477678880, −4.28071552216738719142413551418, −3.15931154897823617709911814459, −1.28545451547438801563788370595, 0,
1.28545451547438801563788370595, 3.15931154897823617709911814459, 4.28071552216738719142413551418, 4.89882587421777491316477678880, 6.13633467934958653813781163805, 7.32444335473090235270531615879, 8.079800066106881357114366719677, 8.421177813158901596650689205879, 10.08943501116698990748926447815
