| L(s) = 1 | − 1.90·2-s + 3-s + 1.63·4-s − 1.90·6-s − 4.45·7-s + 0.689·8-s + 9-s + 1.63·12-s + 0.534·13-s + 8.50·14-s − 4.59·16-s − 4.54·17-s − 1.90·18-s + 1.92·19-s − 4.45·21-s + 5.18·23-s + 0.689·24-s − 1.01·26-s + 27-s − 7.30·28-s − 4.75·29-s + 3.49·31-s + 7.38·32-s + 8.66·34-s + 1.63·36-s + 0.527·37-s − 3.67·38-s + ⋯ |
| L(s) = 1 | − 1.34·2-s + 0.577·3-s + 0.819·4-s − 0.778·6-s − 1.68·7-s + 0.243·8-s + 0.333·9-s + 0.472·12-s + 0.148·13-s + 2.27·14-s − 1.14·16-s − 1.10·17-s − 0.449·18-s + 0.441·19-s − 0.972·21-s + 1.08·23-s + 0.140·24-s − 0.199·26-s + 0.192·27-s − 1.38·28-s − 0.882·29-s + 0.628·31-s + 1.30·32-s + 1.48·34-s + 0.273·36-s + 0.0866·37-s − 0.595·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 1.90T + 2T^{2} \) |
| 7 | \( 1 + 4.45T + 7T^{2} \) |
| 13 | \( 1 - 0.534T + 13T^{2} \) |
| 17 | \( 1 + 4.54T + 17T^{2} \) |
| 19 | \( 1 - 1.92T + 19T^{2} \) |
| 23 | \( 1 - 5.18T + 23T^{2} \) |
| 29 | \( 1 + 4.75T + 29T^{2} \) |
| 31 | \( 1 - 3.49T + 31T^{2} \) |
| 37 | \( 1 - 0.527T + 37T^{2} \) |
| 41 | \( 1 - 4.69T + 41T^{2} \) |
| 43 | \( 1 + 3.02T + 43T^{2} \) |
| 47 | \( 1 + 11.9T + 47T^{2} \) |
| 53 | \( 1 + 5.02T + 53T^{2} \) |
| 59 | \( 1 - 13.6T + 59T^{2} \) |
| 61 | \( 1 - 11.6T + 61T^{2} \) |
| 67 | \( 1 - 11.5T + 67T^{2} \) |
| 71 | \( 1 - 3.36T + 71T^{2} \) |
| 73 | \( 1 - 0.650T + 73T^{2} \) |
| 79 | \( 1 + 17.7T + 79T^{2} \) |
| 83 | \( 1 + 4.24T + 83T^{2} \) |
| 89 | \( 1 + 8.24T + 89T^{2} \) |
| 97 | \( 1 + 1.13T + 97T^{2} \) |
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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
−7.41160868318503004927880087926, −6.86099418705970691394547380849, −6.50291842107402430872679665822, −5.44301867291929619448228843036, −4.44583999830572981478359086539, −3.62268256169790820308108411718, −2.87419766793367859808258135290, −2.11554669678312551650350730790, −0.982303374894578697726586328554, 0,
0.982303374894578697726586328554, 2.11554669678312551650350730790, 2.87419766793367859808258135290, 3.62268256169790820308108411718, 4.44583999830572981478359086539, 5.44301867291929619448228843036, 6.50291842107402430872679665822, 6.86099418705970691394547380849, 7.41160868318503004927880087926
