L(s) = 1 | + 3-s + 5-s + 2.56·7-s + 9-s + 1.43·11-s + 5.12·13-s + 15-s + 1.12·17-s − 1.12·19-s + 2.56·21-s − 4·23-s + 25-s + 27-s + 7.12·29-s + 7.12·31-s + 1.43·33-s + 2.56·35-s − 5.68·37-s + 5.12·39-s + 6·41-s + 2.87·43-s + 45-s − 5.12·47-s − 0.438·49-s + 1.12·51-s + 3.12·53-s + 1.43·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.968·7-s + 0.333·9-s + 0.433·11-s + 1.42·13-s + 0.258·15-s + 0.272·17-s − 0.257·19-s + 0.558·21-s − 0.834·23-s + 0.200·25-s + 0.192·27-s + 1.32·29-s + 1.27·31-s + 0.250·33-s + 0.432·35-s − 0.934·37-s + 0.820·39-s + 0.937·41-s + 0.438·43-s + 0.149·45-s − 0.747·47-s − 0.0626·49-s + 0.157·51-s + 0.428·53-s + 0.193·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.909860017\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.909860017\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 67 | \( 1 - T \) |
good | 7 | \( 1 - 2.56T + 7T^{2} \) |
| 11 | \( 1 - 1.43T + 11T^{2} \) |
| 13 | \( 1 - 5.12T + 13T^{2} \) |
| 17 | \( 1 - 1.12T + 17T^{2} \) |
| 19 | \( 1 + 1.12T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 7.12T + 29T^{2} \) |
| 31 | \( 1 - 7.12T + 31T^{2} \) |
| 37 | \( 1 + 5.68T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 2.87T + 43T^{2} \) |
| 47 | \( 1 + 5.12T + 47T^{2} \) |
| 53 | \( 1 - 3.12T + 53T^{2} \) |
| 59 | \( 1 + 12.2T + 59T^{2} \) |
| 61 | \( 1 - 5.68T + 61T^{2} \) |
| 71 | \( 1 - 7.43T + 71T^{2} \) |
| 73 | \( 1 + 11.1T + 73T^{2} \) |
| 79 | \( 1 + 2T + 79T^{2} \) |
| 83 | \( 1 + 9.43T + 83T^{2} \) |
| 89 | \( 1 + 2.80T + 89T^{2} \) |
| 97 | \( 1 - 14.5T + 97T^{2} \) |
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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
−8.101997652440054771685517138884, −7.19424823207499878875399265354, −6.33997071603033240948723384780, −5.92492918633722996570413706114, −4.88933334625498304352646673370, −4.29508896480258978866115539792, −3.51183954763214654532974670335, −2.62996698887298128459456258773, −1.69509706841606224362470294959, −1.06311274097880722048902186751,
1.06311274097880722048902186751, 1.69509706841606224362470294959, 2.62996698887298128459456258773, 3.51183954763214654532974670335, 4.29508896480258978866115539792, 4.88933334625498304352646673370, 5.92492918633722996570413706114, 6.33997071603033240948723384780, 7.19424823207499878875399265354, 8.101997652440054771685517138884