| L(s) = 1 | − 3-s − 4.02·5-s − 0.910·7-s + 9-s + 5.64·11-s + 6.43·13-s + 4.02·15-s + 3.30·17-s − 1.28·19-s + 0.910·21-s + 9.24·23-s + 11.2·25-s − 27-s + 2.90·29-s + 10.6·31-s − 5.64·33-s + 3.66·35-s + 3.18·37-s − 6.43·39-s + 1.79·41-s − 2.55·43-s − 4.02·45-s − 1.91·47-s − 6.17·49-s − 3.30·51-s − 6.55·53-s − 22.7·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.80·5-s − 0.343·7-s + 0.333·9-s + 1.70·11-s + 1.78·13-s + 1.03·15-s + 0.802·17-s − 0.295·19-s + 0.198·21-s + 1.92·23-s + 2.24·25-s − 0.192·27-s + 0.540·29-s + 1.90·31-s − 0.983·33-s + 0.619·35-s + 0.524·37-s − 1.03·39-s + 0.279·41-s − 0.390·43-s − 0.600·45-s − 0.279·47-s − 0.881·49-s − 0.463·51-s − 0.900·53-s − 3.06·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.637850228\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.637850228\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 + 4.02T + 5T^{2} \) |
| 7 | \( 1 + 0.910T + 7T^{2} \) |
| 11 | \( 1 - 5.64T + 11T^{2} \) |
| 13 | \( 1 - 6.43T + 13T^{2} \) |
| 17 | \( 1 - 3.30T + 17T^{2} \) |
| 19 | \( 1 + 1.28T + 19T^{2} \) |
| 23 | \( 1 - 9.24T + 23T^{2} \) |
| 29 | \( 1 - 2.90T + 29T^{2} \) |
| 31 | \( 1 - 10.6T + 31T^{2} \) |
| 37 | \( 1 - 3.18T + 37T^{2} \) |
| 41 | \( 1 - 1.79T + 41T^{2} \) |
| 43 | \( 1 + 2.55T + 43T^{2} \) |
| 47 | \( 1 + 1.91T + 47T^{2} \) |
| 53 | \( 1 + 6.55T + 53T^{2} \) |
| 59 | \( 1 - 2.97T + 59T^{2} \) |
| 61 | \( 1 - 8.34T + 61T^{2} \) |
| 67 | \( 1 + 4.23T + 67T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 - 1.27T + 73T^{2} \) |
| 79 | \( 1 - 1.79T + 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 - 5.45T + 89T^{2} \) |
| 97 | \( 1 - 13.0T + 97T^{2} \) |
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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
−7.84488278688287963282301900008, −6.98751162528583976527340236093, −6.54078104932147231316975770000, −5.98336344814664128855626780458, −4.74103373230765696226932546421, −4.32251156690414185183710414182, −3.47519622687774994351477622607, −3.17423856880739319758353085848, −1.21320863789813250333520983365, −0.822973061741726542234016974706,
0.822973061741726542234016974706, 1.21320863789813250333520983365, 3.17423856880739319758353085848, 3.47519622687774994351477622607, 4.32251156690414185183710414182, 4.74103373230765696226932546421, 5.98336344814664128855626780458, 6.54078104932147231316975770000, 6.98751162528583976527340236093, 7.84488278688287963282301900008
