| L(s) = 1 | − 11·3-s + 16·4-s + 49·7-s + 40·9-s − 176·12-s + 256·16-s − 466·17-s + 461·19-s − 539·21-s − 769·23-s + 625·25-s + 451·27-s + 784·28-s + 841·29-s + 878·31-s + 640·36-s + 1.01e3·41-s + 4.15e3·47-s − 2.81e3·48-s + 2.40e3·49-s + 5.12e3·51-s + 3.79e3·53-s − 5.07e3·57-s − 1.95e3·61-s + 1.96e3·63-s + 4.09e3·64-s + 7.15e3·67-s + ⋯ |
| L(s) = 1 | − 1.22·3-s + 4-s + 7-s + 0.493·9-s − 1.22·12-s + 16-s − 1.61·17-s + 1.27·19-s − 1.22·21-s − 1.45·23-s + 25-s + 0.618·27-s + 28-s + 29-s + 0.913·31-s + 0.493·36-s + 0.602·41-s + 1.88·47-s − 1.22·48-s + 49-s + 1.97·51-s + 1.34·53-s − 1.56·57-s − 0.525·61-s + 0.493·63-s + 64-s + 1.59·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 203 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 203 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.727244432\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.727244432\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - p^{2} T \) |
| 29 | \( 1 - p^{2} T \) |
| good | 2 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 3 | \( 1 + 11 T + p^{4} T^{2} \) |
| 5 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 11 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 13 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 17 | \( 1 + 466 T + p^{4} T^{2} \) |
| 19 | \( 1 - 461 T + p^{4} T^{2} \) |
| 23 | \( 1 + 769 T + p^{4} T^{2} \) |
| 31 | \( 1 - 878 T + p^{4} T^{2} \) |
| 37 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 41 | \( 1 - 1013 T + p^{4} T^{2} \) |
| 43 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 47 | \( 1 - 4157 T + p^{4} T^{2} \) |
| 53 | \( 1 - 3791 T + p^{4} T^{2} \) |
| 59 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 61 | \( 1 + 1954 T + p^{4} T^{2} \) |
| 67 | \( 1 - 7151 T + p^{4} T^{2} \) |
| 71 | \( 1 + 6361 T + p^{4} T^{2} \) |
| 73 | \( 1 + 10483 T + p^{4} T^{2} \) |
| 79 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 83 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 89 | \( 1 - 15581 T + p^{4} T^{2} \) |
| 97 | \( 1 - 16469 T + p^{4} T^{2} \) |
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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
−11.78714863677171689546668426864, −10.93228150532254278481879223325, −10.30857914710860976359469325384, −8.651413664911564921933711568148, −7.45827389006627207385483015616, −6.48225286060511019161758034360, −5.57013031898467866898216285008, −4.46428968649532843670861620457, −2.46636333175176981229440274490, −0.975312735644753562474984099949,
0.975312735644753562474984099949, 2.46636333175176981229440274490, 4.46428968649532843670861620457, 5.57013031898467866898216285008, 6.48225286060511019161758034360, 7.45827389006627207385483015616, 8.651413664911564921933711568148, 10.30857914710860976359469325384, 10.93228150532254278481879223325, 11.78714863677171689546668426864
