| L(s) = 1 | − 1.06·2-s + 3-s − 0.864·4-s − 5-s − 1.06·6-s − 3.38·7-s + 3.05·8-s + 9-s + 1.06·10-s + 3.71·11-s − 0.864·12-s + 2.48·13-s + 3.60·14-s − 15-s − 1.52·16-s + 6.73·17-s − 1.06·18-s − 7.10·19-s + 0.864·20-s − 3.38·21-s − 3.95·22-s + 3.69·23-s + 3.05·24-s + 25-s − 2.65·26-s + 27-s + 2.92·28-s + ⋯ |
| L(s) = 1 | − 0.753·2-s + 0.577·3-s − 0.432·4-s − 0.447·5-s − 0.435·6-s − 1.27·7-s + 1.07·8-s + 0.333·9-s + 0.336·10-s + 1.12·11-s − 0.249·12-s + 0.690·13-s + 0.964·14-s − 0.258·15-s − 0.381·16-s + 1.63·17-s − 0.251·18-s − 1.62·19-s + 0.193·20-s − 0.738·21-s − 0.844·22-s + 0.770·23-s + 0.623·24-s + 0.200·25-s − 0.520·26-s + 0.192·27-s + 0.552·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.153183154\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.153183154\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 + 1.06T + 2T^{2} \) |
| 7 | \( 1 + 3.38T + 7T^{2} \) |
| 11 | \( 1 - 3.71T + 11T^{2} \) |
| 13 | \( 1 - 2.48T + 13T^{2} \) |
| 17 | \( 1 - 6.73T + 17T^{2} \) |
| 19 | \( 1 + 7.10T + 19T^{2} \) |
| 23 | \( 1 - 3.69T + 23T^{2} \) |
| 29 | \( 1 - 4.52T + 29T^{2} \) |
| 31 | \( 1 + 4.69T + 31T^{2} \) |
| 37 | \( 1 + 8.06T + 37T^{2} \) |
| 41 | \( 1 - 8.96T + 41T^{2} \) |
| 43 | \( 1 - 6.26T + 43T^{2} \) |
| 47 | \( 1 + 7.69T + 47T^{2} \) |
| 53 | \( 1 - 2.83T + 53T^{2} \) |
| 59 | \( 1 - 0.482T + 59T^{2} \) |
| 61 | \( 1 - 2.85T + 61T^{2} \) |
| 67 | \( 1 - 0.558T + 67T^{2} \) |
| 71 | \( 1 - 12.8T + 71T^{2} \) |
| 73 | \( 1 + 8.09T + 73T^{2} \) |
| 79 | \( 1 - 4.36T + 79T^{2} \) |
| 83 | \( 1 + 5.59T + 83T^{2} \) |
| 89 | \( 1 - 12.5T + 89T^{2} \) |
| 97 | \( 1 + 14.7T + 97T^{2} \) |
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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
−8.295220800879484042084199168973, −7.51519011951421302492457508716, −6.80264608367613685265173802609, −6.20725765851365516059139825655, −5.17740332228023386785583835781, −4.02895021041579454549451255025, −3.77374410272172727519547241160, −2.89616305038411296441523053326, −1.56413791068002912175005739825, −0.65784374947387304775494807244,
0.65784374947387304775494807244, 1.56413791068002912175005739825, 2.89616305038411296441523053326, 3.77374410272172727519547241160, 4.02895021041579454549451255025, 5.17740332228023386785583835781, 6.20725765851365516059139825655, 6.80264608367613685265173802609, 7.51519011951421302492457508716, 8.295220800879484042084199168973
