| L(s) = 1 | + 3·3-s + 21.3·5-s − 7·7-s + 9·9-s − 31.3·11-s − 84.7·13-s + 64.0·15-s − 16.0·17-s − 20·19-s − 21·21-s − 80.7·23-s + 331.·25-s + 27·27-s − 102·29-s − 245.·31-s − 94.0·33-s − 149.·35-s − 215.·37-s − 254.·39-s + 150.·41-s − 441.·43-s + 192.·45-s − 206.·47-s + 49·49-s − 48.2·51-s − 426.·53-s − 669.·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.91·5-s − 0.377·7-s + 0.333·9-s − 0.859·11-s − 1.80·13-s + 1.10·15-s − 0.229·17-s − 0.241·19-s − 0.218·21-s − 0.732·23-s + 2.64·25-s + 0.192·27-s − 0.653·29-s − 1.42·31-s − 0.496·33-s − 0.722·35-s − 0.956·37-s − 1.04·39-s + 0.574·41-s − 1.56·43-s + 0.636·45-s − 0.640·47-s + 0.142·49-s − 0.132·51-s − 1.10·53-s − 1.64·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 3T \) |
| 7 | \( 1 + 7T \) |
| good | 5 | \( 1 - 21.3T + 125T^{2} \) |
| 11 | \( 1 + 31.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 84.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 16.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 20T + 6.85e3T^{2} \) |
| 23 | \( 1 + 80.7T + 1.21e4T^{2} \) |
| 29 | \( 1 + 102T + 2.43e4T^{2} \) |
| 31 | \( 1 + 245.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 215.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 150.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 441.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 206.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 426.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 363.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 343.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 69.2T + 3.00e5T^{2} \) |
| 71 | \( 1 + 468.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 747.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.29e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.29e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 563.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.48e3T + 9.12e5T^{2} \) |
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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
−9.179760163346610705098244943679, −8.047601140878086311509698412710, −7.15129402998006261671605592714, −6.37234713996897943516541989660, −5.39131027119174483239366483908, −4.86200205551248960588905930277, −3.30027792610210279411258461193, −2.30356980935062035399057459172, −1.86864537760876230238941328528, 0,
1.86864537760876230238941328528, 2.30356980935062035399057459172, 3.30027792610210279411258461193, 4.86200205551248960588905930277, 5.39131027119174483239366483908, 6.37234713996897943516541989660, 7.15129402998006261671605592714, 8.047601140878086311509698412710, 9.179760163346610705098244943679
