| L(s) = 1 | + 9.21·2-s − 0.734·3-s + 52.9·4-s − 6.76·6-s + 90.0·7-s + 192.·8-s − 242.·9-s − 269.·11-s − 38.8·12-s + 444.·13-s + 829.·14-s + 83.2·16-s − 485.·17-s − 2.23e3·18-s − 1.57e3·19-s − 66.1·21-s − 2.47e3·22-s + 398.·23-s − 141.·24-s + 4.09e3·26-s + 356.·27-s + 4.76e3·28-s − 841·29-s − 8.46e3·31-s − 5.40e3·32-s + 197.·33-s − 4.47e3·34-s + ⋯ |
| L(s) = 1 | + 1.62·2-s − 0.0471·3-s + 1.65·4-s − 0.0767·6-s + 0.694·7-s + 1.06·8-s − 0.997·9-s − 0.670·11-s − 0.0779·12-s + 0.729·13-s + 1.13·14-s + 0.0813·16-s − 0.407·17-s − 1.62·18-s − 0.999·19-s − 0.0327·21-s − 1.09·22-s + 0.157·23-s − 0.0501·24-s + 1.18·26-s + 0.0941·27-s + 1.14·28-s − 0.185·29-s − 1.58·31-s − 0.932·32-s + 0.0315·33-s − 0.663·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 29 | \( 1 + 841T \) |
| good | 2 | \( 1 - 9.21T + 32T^{2} \) |
| 3 | \( 1 + 0.734T + 243T^{2} \) |
| 7 | \( 1 - 90.0T + 1.68e4T^{2} \) |
| 11 | \( 1 + 269.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 444.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 485.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.57e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 398.T + 6.43e6T^{2} \) |
| 31 | \( 1 + 8.46e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 3.33e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.81e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 9.99e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.25e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.13e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.00e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.97e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.75e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.01e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.47e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.84e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.67e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 3.94e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 4.83e4T + 8.58e9T^{2} \) |
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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
−9.022635803810542173229809049591, −8.276004694943540482231025267329, −7.18943929809733559147916217782, −6.13448074240934148726633683476, −5.52402887490294758724875151764, −4.69511982179899889870517049435, −3.76914094846548004057024537084, −2.77367822981141491444928951200, −1.82864606367945013697924106317, 0,
1.82864606367945013697924106317, 2.77367822981141491444928951200, 3.76914094846548004057024537084, 4.69511982179899889870517049435, 5.52402887490294758724875151764, 6.13448074240934148726633683476, 7.18943929809733559147916217782, 8.276004694943540482231025267329, 9.022635803810542173229809049591
