| L(s) = 1 | + 9·3-s − 107.·5-s + 26.6·7-s + 81·9-s − 121·11-s + 904.·13-s − 967.·15-s − 495.·17-s + 1.50e3·19-s + 240.·21-s − 2.39e3·23-s + 8.42e3·25-s + 729·27-s − 5.13e3·29-s + 410.·31-s − 1.08e3·33-s − 2.86e3·35-s + 5.82e3·37-s + 8.14e3·39-s + 1.85e4·41-s + 788.·43-s − 8.70e3·45-s − 3.97e3·47-s − 1.60e4·49-s − 4.45e3·51-s + 1.51e4·53-s + 1.30e4·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.92·5-s + 0.205·7-s + 0.333·9-s − 0.301·11-s + 1.48·13-s − 1.10·15-s − 0.415·17-s + 0.954·19-s + 0.118·21-s − 0.943·23-s + 2.69·25-s + 0.192·27-s − 1.13·29-s + 0.0766·31-s − 0.174·33-s − 0.395·35-s + 0.699·37-s + 0.857·39-s + 1.72·41-s + 0.0650·43-s − 0.640·45-s − 0.262·47-s − 0.957·49-s − 0.240·51-s + 0.739·53-s + 0.579·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - 9T \) |
| 11 | \( 1 + 121T \) |
| good | 5 | \( 1 + 107.T + 3.12e3T^{2} \) |
| 7 | \( 1 - 26.6T + 1.68e4T^{2} \) |
| 13 | \( 1 - 904.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 495.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.50e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.39e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.13e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 410.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.82e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.85e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 788.T + 1.47e8T^{2} \) |
| 47 | \( 1 + 3.97e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.51e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.16e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.99e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.58e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.81e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.38e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.74e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.40e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.58e4T + 5.58e9T^{2} \) |
| 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
−9.404524401038371003844621421276, −8.508453198436435887854783588856, −7.86672593249534010871321628395, −7.28317316181811566040401622183, −5.96545176056967031047866079874, −4.48930055207176550383061119650, −3.82457059821943461645000536626, −2.96169957355391220021879034270, −1.27264775825290167090269437564, 0,
1.27264775825290167090269437564, 2.96169957355391220021879034270, 3.82457059821943461645000536626, 4.48930055207176550383061119650, 5.96545176056967031047866079874, 7.28317316181811566040401622183, 7.86672593249534010871321628395, 8.508453198436435887854783588856, 9.404524401038371003844621421276
