| L(s) = 1 | − 11.1·3-s − 63.3·5-s − 223.·7-s − 118.·9-s + 631.·11-s + 28.5·13-s + 708.·15-s − 1.74e3·17-s + 2.02e3·19-s + 2.49e3·21-s − 2.98e3·23-s + 891.·25-s + 4.03e3·27-s + 766.·29-s + 8.35e3·31-s − 7.06e3·33-s + 1.41e4·35-s + 1.48e4·37-s − 319.·39-s − 5.34e3·41-s + 1.84e3·43-s + 7.47e3·45-s + 6.28e3·47-s + 3.31e4·49-s + 1.94e4·51-s − 915.·53-s − 4.00e4·55-s + ⋯ |
| L(s) = 1 | − 0.717·3-s − 1.13·5-s − 1.72·7-s − 0.485·9-s + 1.57·11-s + 0.0468·13-s + 0.813·15-s − 1.46·17-s + 1.28·19-s + 1.23·21-s − 1.17·23-s + 0.285·25-s + 1.06·27-s + 0.169·29-s + 1.56·31-s − 1.12·33-s + 1.95·35-s + 1.78·37-s − 0.0336·39-s − 0.496·41-s + 0.152·43-s + 0.550·45-s + 0.414·47-s + 1.97·49-s + 1.04·51-s − 0.0447·53-s − 1.78·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 43 | \( 1 - 1.84e3T \) |
| good | 3 | \( 1 + 11.1T + 243T^{2} \) |
| 5 | \( 1 + 63.3T + 3.12e3T^{2} \) |
| 7 | \( 1 + 223.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 631.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 28.5T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.74e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.02e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.98e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 766.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 8.35e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.48e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 5.34e3T + 1.15e8T^{2} \) |
| 47 | \( 1 - 6.28e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 915.T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.46e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.13e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.28e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.64e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.55e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.79e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 7.85e3T + 3.93e9T^{2} \) |
| 89 | \( 1 + 7.56e3T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.11e5T + 8.58e9T^{2} \) |
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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
−9.327662755923625114512104237266, −8.506657654114206950549384660065, −7.32423357771959306364415040204, −6.40833925268135900170324763212, −6.06712381231544913084754952411, −4.48333300611765805403751372603, −3.74329212827487257045124629272, −2.78130757174467428107799188761, −0.839277795865825364333663516562, 0,
0.839277795865825364333663516562, 2.78130757174467428107799188761, 3.74329212827487257045124629272, 4.48333300611765805403751372603, 6.06712381231544913084754952411, 6.40833925268135900170324763212, 7.32423357771959306364415040204, 8.506657654114206950549384660065, 9.327662755923625114512104237266
