| L(s) = 1 | + 5-s − 3.73·7-s + 3.46·11-s + 3.73·13-s + 3.46·17-s + 7.92·19-s − 0.464·23-s + 25-s − 6.92·29-s − 5.46·31-s − 3.73·35-s + 8·37-s + 5.19·41-s + 1.46·43-s + 6.46·47-s + 6.92·49-s − 12.4·53-s + 3.46·55-s − 8.66·59-s − 8.39·61-s + 3.73·65-s − 8·67-s + 6·71-s + 6.39·73-s − 12.9·77-s − 6.39·79-s + 8.53·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.41·7-s + 1.04·11-s + 1.03·13-s + 0.840·17-s + 1.81·19-s − 0.0967·23-s + 0.200·25-s − 1.28·29-s − 0.981·31-s − 0.630·35-s + 1.31·37-s + 0.811·41-s + 0.223·43-s + 0.942·47-s + 0.989·49-s − 1.71·53-s + 0.467·55-s − 1.12·59-s − 1.07·61-s + 0.462·65-s − 0.977·67-s + 0.712·71-s + 0.748·73-s − 1.47·77-s − 0.719·79-s + 0.936·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.224368756\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.224368756\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| good | 7 | \( 1 + 3.73T + 7T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 - 3.73T + 13T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 19 | \( 1 - 7.92T + 19T^{2} \) |
| 23 | \( 1 + 0.464T + 23T^{2} \) |
| 29 | \( 1 + 6.92T + 29T^{2} \) |
| 31 | \( 1 + 5.46T + 31T^{2} \) |
| 37 | \( 1 - 8T + 37T^{2} \) |
| 41 | \( 1 - 5.19T + 41T^{2} \) |
| 43 | \( 1 - 1.46T + 43T^{2} \) |
| 47 | \( 1 - 6.46T + 47T^{2} \) |
| 53 | \( 1 + 12.4T + 53T^{2} \) |
| 59 | \( 1 + 8.66T + 59T^{2} \) |
| 61 | \( 1 + 8.39T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 - 6.39T + 73T^{2} \) |
| 79 | \( 1 + 6.39T + 79T^{2} \) |
| 83 | \( 1 - 8.53T + 83T^{2} \) |
| 89 | \( 1 - 12T + 89T^{2} \) |
| 97 | \( 1 - 1.07T + 97T^{2} \) |
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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
−7.81954393643078066454034022857, −7.34931342603331488401936437746, −6.41697480664872051245282984620, −5.99243939984286688903532971375, −5.43431451567671090978770380573, −4.21796911302626611004598862950, −3.43358109148721695463216454045, −3.05528812677609273491182678646, −1.68679312723488129557104913657, −0.813472835380244922495323187130,
0.813472835380244922495323187130, 1.68679312723488129557104913657, 3.05528812677609273491182678646, 3.43358109148721695463216454045, 4.21796911302626611004598862950, 5.43431451567671090978770380573, 5.99243939984286688903532971375, 6.41697480664872051245282984620, 7.34931342603331488401936437746, 7.81954393643078066454034022857
