| L(s) = 1 | + 3-s − 5-s + 2·7-s + 9-s − 4·13-s − 15-s − 3·17-s − 4·19-s + 2·21-s − 3·23-s + 25-s + 27-s − 31-s − 2·35-s + 2·37-s − 4·39-s + 6·41-s + 8·43-s − 45-s + 3·47-s − 3·49-s − 3·51-s + 9·53-s − 4·57-s + 12·59-s + 5·61-s + 2·63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s − 1.10·13-s − 0.258·15-s − 0.727·17-s − 0.917·19-s + 0.436·21-s − 0.625·23-s + 1/5·25-s + 0.192·27-s − 0.179·31-s − 0.338·35-s + 0.328·37-s − 0.640·39-s + 0.937·41-s + 1.21·43-s − 0.149·45-s + 0.437·47-s − 3/7·49-s − 0.420·51-s + 1.23·53-s − 0.529·57-s + 1.56·59-s + 0.640·61-s + 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.124867252\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.124867252\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
| 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
−7.945001307084333761622895115269, −7.34727381238601111923277239189, −6.72101202589549679011987477491, −5.78336989384541804481276693834, −4.90065488878689926500735507325, −4.30596859267544144522564468789, −3.69586595894868074193249808778, −2.43209498910133906644558003272, −2.13248802979161633324636611024, −0.69617127176433802718022892925,
0.69617127176433802718022892925, 2.13248802979161633324636611024, 2.43209498910133906644558003272, 3.69586595894868074193249808778, 4.30596859267544144522564468789, 4.90065488878689926500735507325, 5.78336989384541804481276693834, 6.72101202589549679011987477491, 7.34727381238601111923277239189, 7.945001307084333761622895115269
