| L(s) = 1 | − 1.00·2-s − 1.30·3-s − 0.981·4-s + 5-s + 1.31·6-s + 1.28·7-s + 3.00·8-s − 1.29·9-s − 1.00·10-s − 11-s + 1.28·12-s + 3.40·13-s − 1.29·14-s − 1.30·15-s − 1.07·16-s + 0.582·17-s + 1.31·18-s − 8.39·19-s − 0.981·20-s − 1.67·21-s + 1.00·22-s + 2.16·23-s − 3.92·24-s + 25-s − 3.43·26-s + 5.60·27-s − 1.25·28-s + ⋯ |
| L(s) = 1 | − 0.713·2-s − 0.753·3-s − 0.490·4-s + 0.447·5-s + 0.537·6-s + 0.484·7-s + 1.06·8-s − 0.432·9-s − 0.319·10-s − 0.301·11-s + 0.369·12-s + 0.943·13-s − 0.345·14-s − 0.336·15-s − 0.268·16-s + 0.141·17-s + 0.308·18-s − 1.92·19-s − 0.219·20-s − 0.364·21-s + 0.215·22-s + 0.452·23-s − 0.801·24-s + 0.200·25-s − 0.672·26-s + 1.07·27-s − 0.237·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7513751423\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7513751423\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 + T \) |
| good | 2 | \( 1 + 1.00T + 2T^{2} \) |
| 3 | \( 1 + 1.30T + 3T^{2} \) |
| 7 | \( 1 - 1.28T + 7T^{2} \) |
| 13 | \( 1 - 3.40T + 13T^{2} \) |
| 17 | \( 1 - 0.582T + 17T^{2} \) |
| 19 | \( 1 + 8.39T + 19T^{2} \) |
| 23 | \( 1 - 2.16T + 23T^{2} \) |
| 29 | \( 1 + 8.58T + 29T^{2} \) |
| 31 | \( 1 - 6.13T + 31T^{2} \) |
| 37 | \( 1 - 9.14T + 37T^{2} \) |
| 41 | \( 1 - 12.0T + 41T^{2} \) |
| 43 | \( 1 - 1.77T + 43T^{2} \) |
| 47 | \( 1 - 1.55T + 47T^{2} \) |
| 53 | \( 1 - 0.0890T + 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 + 6.57T + 61T^{2} \) |
| 67 | \( 1 + 12.0T + 67T^{2} \) |
| 71 | \( 1 + 10.2T + 71T^{2} \) |
| 79 | \( 1 + 8.40T + 79T^{2} \) |
| 83 | \( 1 + 2.27T + 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 - 6.16T + 97T^{2} \) |
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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
−8.491522444777252660312144355485, −7.966243600994782451151654980761, −7.03983933475856766493158079024, −5.97354041042395091921792542633, −5.75943639455705644352449241773, −4.61651890601667474168698193704, −4.17565822111842149454947503795, −2.75339421568266136550580548202, −1.63743561671757509045971209962, −0.59892215593583041491902273395,
0.59892215593583041491902273395, 1.63743561671757509045971209962, 2.75339421568266136550580548202, 4.17565822111842149454947503795, 4.61651890601667474168698193704, 5.75943639455705644352449241773, 5.97354041042395091921792542633, 7.03983933475856766493158079024, 7.966243600994782451151654980761, 8.491522444777252660312144355485
