| L(s) = 1 | + 5-s + 7-s + 0.627·11-s − 1.37·13-s − 5.37·17-s − 6.74·19-s + 6.74·23-s + 25-s − 1.37·29-s + 8·31-s + 35-s − 2·37-s + 4.74·41-s − 2.74·43-s + 10.1·47-s + 49-s + 0.744·53-s + 0.627·55-s + 8·59-s + 8.74·61-s − 1.37·65-s + 4·67-s + 8·71-s − 6·73-s + 0.627·77-s + 2.11·79-s + 13.4·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.377·7-s + 0.189·11-s − 0.380·13-s − 1.30·17-s − 1.54·19-s + 1.40·23-s + 0.200·25-s − 0.254·29-s + 1.43·31-s + 0.169·35-s − 0.328·37-s + 0.740·41-s − 0.418·43-s + 1.47·47-s + 0.142·49-s + 0.102·53-s + 0.0846·55-s + 1.04·59-s + 1.11·61-s − 0.170·65-s + 0.488·67-s + 0.949·71-s − 0.702·73-s + 0.0715·77-s + 0.238·79-s + 1.48·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.069261112\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.069261112\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 - 0.627T + 11T^{2} \) |
| 13 | \( 1 + 1.37T + 13T^{2} \) |
| 17 | \( 1 + 5.37T + 17T^{2} \) |
| 19 | \( 1 + 6.74T + 19T^{2} \) |
| 23 | \( 1 - 6.74T + 23T^{2} \) |
| 29 | \( 1 + 1.37T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 4.74T + 41T^{2} \) |
| 43 | \( 1 + 2.74T + 43T^{2} \) |
| 47 | \( 1 - 10.1T + 47T^{2} \) |
| 53 | \( 1 - 0.744T + 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 - 8.74T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 - 2.11T + 79T^{2} \) |
| 83 | \( 1 - 13.4T + 83T^{2} \) |
| 89 | \( 1 + 3.25T + 89T^{2} \) |
| 97 | \( 1 - 18.8T + 97T^{2} \) |
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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
−8.453027180240026990238995733635, −7.43701935821757984501527045080, −6.70767752794811460316172895904, −6.23291836127871308536914088935, −5.20550267824317913418680455933, −4.60648272386614605331159034846, −3.86019546216161487919496058929, −2.59824069873603658727061867360, −2.07471268007318952461937521987, −0.78000202335611765641632846729,
0.78000202335611765641632846729, 2.07471268007318952461937521987, 2.59824069873603658727061867360, 3.86019546216161487919496058929, 4.60648272386614605331159034846, 5.20550267824317913418680455933, 6.23291836127871308536914088935, 6.70767752794811460316172895904, 7.43701935821757984501527045080, 8.453027180240026990238995733635
