| L(s) = 1 | + 5·5-s + 36·7-s − 64·11-s − 65·13-s − 59·17-s − 28·19-s − 160·23-s − 100·25-s + 57·29-s + 164·31-s + 180·35-s − 321·37-s + 246·41-s − 8·43-s − 84·47-s + 953·49-s − 478·53-s − 320·55-s + 32·59-s + 415·61-s − 325·65-s − 220·67-s − 884·71-s − 77·73-s − 2.30e3·77-s − 80·79-s − 1.26e3·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.94·7-s − 1.75·11-s − 1.38·13-s − 0.841·17-s − 0.338·19-s − 1.45·23-s − 4/5·25-s + 0.364·29-s + 0.950·31-s + 0.869·35-s − 1.42·37-s + 0.937·41-s − 0.0283·43-s − 0.260·47-s + 2.77·49-s − 1.23·53-s − 0.784·55-s + 0.0706·59-s + 0.871·61-s − 0.620·65-s − 0.401·67-s − 1.47·71-s − 0.123·73-s − 3.40·77-s − 0.113·79-s − 1.67·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - p T + p^{3} T^{2} \) |
| 7 | \( 1 - 36 T + p^{3} T^{2} \) |
| 11 | \( 1 + 64 T + p^{3} T^{2} \) |
| 13 | \( 1 + 5 p T + p^{3} T^{2} \) |
| 17 | \( 1 + 59 T + p^{3} T^{2} \) |
| 19 | \( 1 + 28 T + p^{3} T^{2} \) |
| 23 | \( 1 + 160 T + p^{3} T^{2} \) |
| 29 | \( 1 - 57 T + p^{3} T^{2} \) |
| 31 | \( 1 - 164 T + p^{3} T^{2} \) |
| 37 | \( 1 + 321 T + p^{3} T^{2} \) |
| 41 | \( 1 - 6 p T + p^{3} T^{2} \) |
| 43 | \( 1 + 8 T + p^{3} T^{2} \) |
| 47 | \( 1 + 84 T + p^{3} T^{2} \) |
| 53 | \( 1 + 478 T + p^{3} T^{2} \) |
| 59 | \( 1 - 32 T + p^{3} T^{2} \) |
| 61 | \( 1 - 415 T + p^{3} T^{2} \) |
| 67 | \( 1 + 220 T + p^{3} T^{2} \) |
| 71 | \( 1 + 884 T + p^{3} T^{2} \) |
| 73 | \( 1 + 77 T + p^{3} T^{2} \) |
| 79 | \( 1 + 80 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1268 T + p^{3} T^{2} \) |
| 89 | \( 1 + 123 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1346 T + p^{3} T^{2} \) |
| 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
−9.989514777843461467606164984600, −8.634191585719677580572840993092, −7.949375464588405596191014982636, −7.35841264185730459007732426255, −5.88357572625680218902725130840, −5.00196768449175476595257977339, −4.45372952638966077046282947168, −2.49864291636430766114605436375, −1.87960814262257705537372763146, 0,
1.87960814262257705537372763146, 2.49864291636430766114605436375, 4.45372952638966077046282947168, 5.00196768449175476595257977339, 5.88357572625680218902725130840, 7.35841264185730459007732426255, 7.949375464588405596191014982636, 8.634191585719677580572840993092, 9.989514777843461467606164984600
