| L(s) = 1 | + 31.5·3-s − 625·5-s − 5.76e3·7-s − 1.86e4·9-s + 4.00e4·11-s − 6.68e4·13-s − 1.97e4·15-s − 9.14e4·17-s − 9.01e5·19-s − 1.81e5·21-s + 2.60e5·23-s + 3.90e5·25-s − 1.21e6·27-s − 8.22e5·29-s + 8.24e6·31-s + 1.26e6·33-s + 3.60e6·35-s − 3.26e6·37-s − 2.10e6·39-s − 7.21e5·41-s + 4.71e6·43-s + 1.16e7·45-s + 1.54e6·47-s − 7.14e6·49-s − 2.88e6·51-s + 4.95e7·53-s − 2.50e7·55-s + ⋯ |
| L(s) = 1 | + 0.224·3-s − 0.447·5-s − 0.907·7-s − 0.949·9-s + 0.825·11-s − 0.649·13-s − 0.100·15-s − 0.265·17-s − 1.58·19-s − 0.203·21-s + 0.194·23-s + 0.200·25-s − 0.438·27-s − 0.215·29-s + 1.60·31-s + 0.185·33-s + 0.405·35-s − 0.286·37-s − 0.146·39-s − 0.0398·41-s + 0.210·43-s + 0.424·45-s + 0.0461·47-s − 0.176·49-s − 0.0596·51-s + 0.863·53-s − 0.369·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.106663802\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.106663802\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 625T \) |
| good | 3 | \( 1 - 31.5T + 1.96e4T^{2} \) |
| 7 | \( 1 + 5.76e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 4.00e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 6.68e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 9.14e4T + 1.18e11T^{2} \) |
| 19 | \( 1 + 9.01e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 2.60e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 8.22e5T + 1.45e13T^{2} \) |
| 31 | \( 1 - 8.24e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 3.26e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 7.21e5T + 3.27e14T^{2} \) |
| 43 | \( 1 - 4.71e6T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.54e6T + 1.11e15T^{2} \) |
| 53 | \( 1 - 4.95e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.47e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.31e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 3.13e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.43e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 7.49e6T + 5.88e16T^{2} \) |
| 79 | \( 1 + 4.72e7T + 1.19e17T^{2} \) |
| 83 | \( 1 + 5.06e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 2.62e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.33e9T + 7.60e17T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.27883060355314891695923840816, −10.11779267225740409691090766588, −9.056706042944545503621225079872, −8.264286179802283771504684693988, −6.89608398285026782050129488471, −6.06504075406218904295988305688, −4.52537438100135420988563785053, −3.38653178912414740939446446388, −2.29903497738909627219863563559, −0.49767353109249724893970600427,
0.49767353109249724893970600427, 2.29903497738909627219863563559, 3.38653178912414740939446446388, 4.52537438100135420988563785053, 6.06504075406218904295988305688, 6.89608398285026782050129488471, 8.264286179802283771504684693988, 9.056706042944545503621225079872, 10.11779267225740409691090766588, 11.27883060355314891695923840816
