| L(s) = 1 | − 3-s − 5-s + 2·7-s + 9-s + 2·11-s + 15-s − 17-s + 6·19-s − 2·21-s + 2·23-s + 25-s − 27-s + 3·29-s − 31-s − 2·33-s − 2·35-s + 8·37-s − 10·41-s + 2·43-s − 45-s − 8·47-s − 3·49-s + 51-s + 13·53-s − 2·55-s − 6·57-s − 13·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s + 0.603·11-s + 0.258·15-s − 0.242·17-s + 1.37·19-s − 0.436·21-s + 0.417·23-s + 1/5·25-s − 0.192·27-s + 0.557·29-s − 0.179·31-s − 0.348·33-s − 0.338·35-s + 1.31·37-s − 1.56·41-s + 0.304·43-s − 0.149·45-s − 1.16·47-s − 3/7·49-s + 0.140·51-s + 1.78·53-s − 0.269·55-s − 0.794·57-s − 1.69·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 344760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 344760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 13 T + p T^{2} \) |
| 59 | \( 1 + 13 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 11 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
| 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
−12.88714570086247, −12.02370017731662, −11.84803896823471, −11.52207950440824, −11.15025232403531, −10.50302083431575, −10.25105752049657, −9.519026318149695, −9.213769655859592, −8.696704886482226, −8.062248204095957, −7.751928473778177, −7.301549609538187, −6.645922746076240, −6.434884347398190, −5.697169943501218, −5.199450913895963, −4.834278052951408, −4.335833344060197, −3.818097269265891, −3.180876969488784, −2.729737315784915, −1.821460017344983, −1.338307676033137, −0.8323018678802036, 0,
0.8323018678802036, 1.338307676033137, 1.821460017344983, 2.729737315784915, 3.180876969488784, 3.818097269265891, 4.335833344060197, 4.834278052951408, 5.199450913895963, 5.697169943501218, 6.434884347398190, 6.645922746076240, 7.301549609538187, 7.751928473778177, 8.062248204095957, 8.696704886482226, 9.213769655859592, 9.519026318149695, 10.25105752049657, 10.50302083431575, 11.15025232403531, 11.52207950440824, 11.84803896823471, 12.02370017731662, 12.88714570086247
