| L(s) = 1 | − 3-s + 5-s + 7-s + 9-s + 6·13-s − 15-s − 17-s − 4·19-s − 21-s + 25-s − 27-s + 2·29-s + 35-s + 6·37-s − 6·39-s − 10·41-s − 4·43-s + 45-s + 4·47-s + 49-s + 51-s − 6·53-s + 4·57-s − 4·59-s + 14·61-s + 63-s + 6·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s + 1.66·13-s − 0.258·15-s − 0.242·17-s − 0.917·19-s − 0.218·21-s + 1/5·25-s − 0.192·27-s + 0.371·29-s + 0.169·35-s + 0.986·37-s − 0.960·39-s − 1.56·41-s − 0.609·43-s + 0.149·45-s + 0.583·47-s + 1/7·49-s + 0.140·51-s − 0.824·53-s + 0.529·57-s − 0.520·59-s + 1.79·61-s + 0.125·63-s + 0.744·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
| 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
−13.86584734386824, −13.27218479058830, −12.92564954689682, −12.53191801749068, −11.78227338736297, −11.24733145531632, −11.14760068348048, −10.48525653687245, −10.02894121859562, −9.581414849165790, −8.765638668732336, −8.476891136679326, −8.114224414862752, −7.263047212061065, −6.654568401764810, −6.403298734510624, −5.771049308527858, −5.369623821458513, −4.690291218885432, −4.149302281035464, −3.649839235303945, −2.886260064584906, −2.137523034445710, −1.501984451508579, −0.9746839047957031, 0,
0.9746839047957031, 1.501984451508579, 2.137523034445710, 2.886260064584906, 3.649839235303945, 4.149302281035464, 4.690291218885432, 5.369623821458513, 5.771049308527858, 6.403298734510624, 6.654568401764810, 7.263047212061065, 8.114224414862752, 8.476891136679326, 8.765638668732336, 9.581414849165790, 10.02894121859562, 10.48525653687245, 11.14760068348048, 11.24733145531632, 11.78227338736297, 12.53191801749068, 12.92564954689682, 13.27218479058830, 13.86584734386824
