| L(s) = 1 | − 5-s − 4.76·7-s − 5.60·11-s + 3.60·13-s + 4.12·17-s + 2.64·19-s + 23-s + 25-s − 3.87·29-s + 9.79·31-s + 4.76·35-s − 6.76·37-s + 11.3·41-s − 2.96·43-s − 1.35·47-s + 15.7·49-s + 3.09·53-s + 5.60·55-s − 12.4·59-s − 10.5·61-s − 3.60·65-s + 6.82·67-s − 11.6·71-s + 10.1·73-s + 26.7·77-s + 12.4·79-s − 10.1·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.80·7-s − 1.69·11-s + 1.00·13-s + 1.00·17-s + 0.605·19-s + 0.208·23-s + 0.200·25-s − 0.719·29-s + 1.75·31-s + 0.805·35-s − 1.11·37-s + 1.77·41-s − 0.452·43-s − 0.198·47-s + 2.24·49-s + 0.425·53-s + 0.756·55-s − 1.61·59-s − 1.35·61-s − 0.447·65-s + 0.833·67-s − 1.38·71-s + 1.19·73-s + 3.04·77-s + 1.40·79-s − 1.11·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 + 4.76T + 7T^{2} \) |
| 11 | \( 1 + 5.60T + 11T^{2} \) |
| 13 | \( 1 - 3.60T + 13T^{2} \) |
| 17 | \( 1 - 4.12T + 17T^{2} \) |
| 19 | \( 1 - 2.64T + 19T^{2} \) |
| 29 | \( 1 + 3.87T + 29T^{2} \) |
| 31 | \( 1 - 9.79T + 31T^{2} \) |
| 37 | \( 1 + 6.76T + 37T^{2} \) |
| 41 | \( 1 - 11.3T + 41T^{2} \) |
| 43 | \( 1 + 2.96T + 43T^{2} \) |
| 47 | \( 1 + 1.35T + 47T^{2} \) |
| 53 | \( 1 - 3.09T + 53T^{2} \) |
| 59 | \( 1 + 12.4T + 59T^{2} \) |
| 61 | \( 1 + 10.5T + 61T^{2} \) |
| 67 | \( 1 - 6.82T + 67T^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 73 | \( 1 - 10.1T + 73T^{2} \) |
| 79 | \( 1 - 12.4T + 79T^{2} \) |
| 83 | \( 1 + 10.1T + 83T^{2} \) |
| 89 | \( 1 - 8.31T + 89T^{2} \) |
| 97 | \( 1 - 16.8T + 97T^{2} \) |
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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
−7.63939691581110607432257178021, −6.72405414235079632006026881196, −6.06858980254164815885640140534, −5.52637499495720255173116106744, −4.66305983279904894740343488559, −3.59002786662632905831139635686, −3.19607220065323497367424238121, −2.52333502799017945153129169672, −1.00224535463596212962544504742, 0,
1.00224535463596212962544504742, 2.52333502799017945153129169672, 3.19607220065323497367424238121, 3.59002786662632905831139635686, 4.66305983279904894740343488559, 5.52637499495720255173116106744, 6.06858980254164815885640140534, 6.72405414235079632006026881196, 7.63939691581110607432257178021
