| L(s) = 1 | − 2.45·2-s − 2.50·3-s + 4.01·4-s − 3.16·5-s + 6.13·6-s − 0.556·7-s − 4.93·8-s + 3.25·9-s + 7.75·10-s + 0.948·11-s − 10.0·12-s − 4.46·13-s + 1.36·14-s + 7.90·15-s + 4.08·16-s + 0.621·17-s − 7.97·18-s + 3.59·19-s − 12.6·20-s + 1.39·21-s − 2.32·22-s + 4.39·23-s + 12.3·24-s + 5.00·25-s + 10.9·26-s − 0.625·27-s − 2.23·28-s + ⋯ |
| L(s) = 1 | − 1.73·2-s − 1.44·3-s + 2.00·4-s − 1.41·5-s + 2.50·6-s − 0.210·7-s − 1.74·8-s + 1.08·9-s + 2.45·10-s + 0.285·11-s − 2.89·12-s − 1.23·13-s + 0.364·14-s + 2.04·15-s + 1.02·16-s + 0.150·17-s − 1.87·18-s + 0.824·19-s − 2.83·20-s + 0.303·21-s − 0.495·22-s + 0.915·23-s + 2.52·24-s + 1.00·25-s + 2.14·26-s − 0.120·27-s − 0.421·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6031 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6031 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2314936864\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2314936864\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + T \) |
| 163 | \( 1 - T \) |
| good | 2 | \( 1 + 2.45T + 2T^{2} \) |
| 3 | \( 1 + 2.50T + 3T^{2} \) |
| 5 | \( 1 + 3.16T + 5T^{2} \) |
| 7 | \( 1 + 0.556T + 7T^{2} \) |
| 11 | \( 1 - 0.948T + 11T^{2} \) |
| 13 | \( 1 + 4.46T + 13T^{2} \) |
| 17 | \( 1 - 0.621T + 17T^{2} \) |
| 19 | \( 1 - 3.59T + 19T^{2} \) |
| 23 | \( 1 - 4.39T + 23T^{2} \) |
| 29 | \( 1 - 0.608T + 29T^{2} \) |
| 31 | \( 1 - 5.87T + 31T^{2} \) |
| 41 | \( 1 - 4.96T + 41T^{2} \) |
| 43 | \( 1 + 12.0T + 43T^{2} \) |
| 47 | \( 1 - 9.02T + 47T^{2} \) |
| 53 | \( 1 - 0.333T + 53T^{2} \) |
| 59 | \( 1 - 4.14T + 59T^{2} \) |
| 61 | \( 1 - 3.73T + 61T^{2} \) |
| 67 | \( 1 + 0.200T + 67T^{2} \) |
| 71 | \( 1 - 16.0T + 71T^{2} \) |
| 73 | \( 1 - 6.90T + 73T^{2} \) |
| 79 | \( 1 - 16.1T + 79T^{2} \) |
| 83 | \( 1 + 4.01T + 83T^{2} \) |
| 89 | \( 1 + 9.80T + 89T^{2} \) |
| 97 | \( 1 + 9.24T + 97T^{2} \) |
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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
−8.029517305571224692948394779941, −7.39456842270134220992119978557, −6.91250306769432242739253217389, −6.36198320764205402513506741424, −5.23044533270761986874945423507, −4.66980776790479755870070058350, −3.53426697064811620631346342675, −2.51622980517440114243868548871, −1.09114483718217721160893996349, −0.43999049776889195655186626443,
0.43999049776889195655186626443, 1.09114483718217721160893996349, 2.51622980517440114243868548871, 3.53426697064811620631346342675, 4.66980776790479755870070058350, 5.23044533270761986874945423507, 6.36198320764205402513506741424, 6.91250306769432242739253217389, 7.39456842270134220992119978557, 8.029517305571224692948394779941
