| L(s) = 1 | + 3-s + 5-s + 9-s + 13-s + 15-s − 2·19-s + 2·23-s + 25-s + 27-s + 10·29-s − 2·31-s + 39-s + 2·41-s + 4·43-s + 45-s + 4·47-s − 7·49-s − 2·57-s − 12·59-s − 14·61-s + 65-s + 2·69-s + 4·73-s + 75-s + 8·79-s + 81-s + 4·83-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1/3·9-s + 0.277·13-s + 0.258·15-s − 0.458·19-s + 0.417·23-s + 1/5·25-s + 0.192·27-s + 1.85·29-s − 0.359·31-s + 0.160·39-s + 0.312·41-s + 0.609·43-s + 0.149·45-s + 0.583·47-s − 49-s − 0.264·57-s − 1.56·59-s − 1.79·61-s + 0.124·65-s + 0.240·69-s + 0.468·73-s + 0.115·75-s + 0.900·79-s + 1/9·81-s + 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.031193609\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.031193609\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.14829925024955, −12.66687942884738, −12.29201566187528, −11.82184372873168, −11.07304328396618, −10.71659322560776, −10.32387587444885, −9.779612544067278, −9.140169385841797, −9.010696317329591, −8.381042865171400, −7.841828966666467, −7.498716717359827, −6.744265382099149, −6.355310459375224, −5.962418409618859, −5.203757657007481, −4.647452066815893, −4.323376440811435, −3.471408282357457, −3.078339605839900, −2.483455665803458, −1.892953993367161, −1.255431794596829, −0.5644117361778444,
0.5644117361778444, 1.255431794596829, 1.892953993367161, 2.483455665803458, 3.078339605839900, 3.471408282357457, 4.323376440811435, 4.647452066815893, 5.203757657007481, 5.962418409618859, 6.355310459375224, 6.744265382099149, 7.498716717359827, 7.841828966666467, 8.381042865171400, 9.010696317329591, 9.140169385841797, 9.779612544067278, 10.32387587444885, 10.71659322560776, 11.07304328396618, 11.82184372873168, 12.29201566187528, 12.66687942884738, 13.14829925024955
