| L(s) = 1 | − 2.32·2-s + 2.99·3-s + 3.38·4-s + 5-s − 6.95·6-s − 1.69·7-s − 3.21·8-s + 5.98·9-s − 2.32·10-s − 6.35·11-s + 10.1·12-s + 4.05·13-s + 3.94·14-s + 2.99·15-s + 0.682·16-s − 4.59·17-s − 13.8·18-s − 6.05·19-s + 3.38·20-s − 5.09·21-s + 14.7·22-s − 7.29·23-s − 9.62·24-s + 25-s − 9.40·26-s + 8.93·27-s − 5.74·28-s + ⋯ |
| L(s) = 1 | − 1.64·2-s + 1.73·3-s + 1.69·4-s + 0.447·5-s − 2.83·6-s − 0.642·7-s − 1.13·8-s + 1.99·9-s − 0.733·10-s − 1.91·11-s + 2.92·12-s + 1.12·13-s + 1.05·14-s + 0.773·15-s + 0.170·16-s − 1.11·17-s − 3.27·18-s − 1.38·19-s + 0.756·20-s − 1.11·21-s + 3.14·22-s − 1.52·23-s − 1.96·24-s + 0.200·25-s − 1.84·26-s + 1.71·27-s − 1.08·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 + 2.32T + 2T^{2} \) |
| 3 | \( 1 - 2.99T + 3T^{2} \) |
| 7 | \( 1 + 1.69T + 7T^{2} \) |
| 11 | \( 1 + 6.35T + 11T^{2} \) |
| 13 | \( 1 - 4.05T + 13T^{2} \) |
| 17 | \( 1 + 4.59T + 17T^{2} \) |
| 19 | \( 1 + 6.05T + 19T^{2} \) |
| 23 | \( 1 + 7.29T + 23T^{2} \) |
| 29 | \( 1 + 7.99T + 29T^{2} \) |
| 31 | \( 1 + 0.190T + 31T^{2} \) |
| 37 | \( 1 - 4.85T + 37T^{2} \) |
| 41 | \( 1 - 5.12T + 41T^{2} \) |
| 43 | \( 1 + 4.94T + 43T^{2} \) |
| 47 | \( 1 + 4.43T + 47T^{2} \) |
| 53 | \( 1 + 11.5T + 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 + 7.55T + 61T^{2} \) |
| 67 | \( 1 - 1.56T + 67T^{2} \) |
| 71 | \( 1 - 7.06T + 71T^{2} \) |
| 73 | \( 1 + 0.138T + 73T^{2} \) |
| 79 | \( 1 - 2.40T + 79T^{2} \) |
| 83 | \( 1 + 9.39T + 83T^{2} \) |
| 89 | \( 1 - 17.5T + 89T^{2} \) |
| 97 | \( 1 - 2.91T + 97T^{2} \) |
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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
−8.722739649105873020076015025447, −8.120281960355252791662162588025, −7.80737517822189197750842729573, −6.76719799446969427298262501661, −6.00935571442464118414747607669, −4.40555750356936236213490760672, −3.30161583098416036980360020999, −2.29033844830519984863396911503, −1.92295550265841882592465222991, 0,
1.92295550265841882592465222991, 2.29033844830519984863396911503, 3.30161583098416036980360020999, 4.40555750356936236213490760672, 6.00935571442464118414747607669, 6.76719799446969427298262501661, 7.80737517822189197750842729573, 8.120281960355252791662162588025, 8.722739649105873020076015025447
