L(s) = 1 | + 2·3-s + 5-s − 4·7-s + 9-s − 4·11-s − 4·13-s + 2·15-s − 2·17-s + 19-s − 8·21-s + 25-s − 4·27-s + 2·29-s − 8·31-s − 8·33-s − 4·35-s + 4·37-s − 8·39-s + 6·41-s + 45-s − 12·47-s + 9·49-s − 4·51-s − 8·53-s − 4·55-s + 2·57-s + 2·61-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 0.447·5-s − 1.51·7-s + 1/3·9-s − 1.20·11-s − 1.10·13-s + 0.516·15-s − 0.485·17-s + 0.229·19-s − 1.74·21-s + 1/5·25-s − 0.769·27-s + 0.371·29-s − 1.43·31-s − 1.39·33-s − 0.676·35-s + 0.657·37-s − 1.28·39-s + 0.937·41-s + 0.149·45-s − 1.75·47-s + 9/7·49-s − 0.560·51-s − 1.09·53-s − 0.539·55-s + 0.264·57-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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
−9.378510991941256926320599037402, −8.275934450088895581467796067311, −7.56186383761303726301658586747, −6.75748996467427670181333689729, −5.81913226127580309415137377006, −4.88072322821672453742216928381, −3.58012982806994395167456994162, −2.82540491064399701707550154132, −2.21004067490187676234569474220, 0,
2.21004067490187676234569474220, 2.82540491064399701707550154132, 3.58012982806994395167456994162, 4.88072322821672453742216928381, 5.81913226127580309415137377006, 6.75748996467427670181333689729, 7.56186383761303726301658586747, 8.275934450088895581467796067311, 9.378510991941256926320599037402