| L(s) = 1 | − 2.62·2-s + 3-s + 4.89·4-s + 5-s − 2.62·6-s − 0.417·7-s − 7.61·8-s + 9-s − 2.62·10-s + 0.944·11-s + 4.89·12-s + 0.996·13-s + 1.09·14-s + 15-s + 10.1·16-s − 1.75·17-s − 2.62·18-s − 8.64·19-s + 4.89·20-s − 0.417·21-s − 2.48·22-s + 5.41·23-s − 7.61·24-s + 25-s − 2.61·26-s + 27-s − 2.04·28-s + ⋯ |
| L(s) = 1 | − 1.85·2-s + 0.577·3-s + 2.44·4-s + 0.447·5-s − 1.07·6-s − 0.157·7-s − 2.69·8-s + 0.333·9-s − 0.830·10-s + 0.284·11-s + 1.41·12-s + 0.276·13-s + 0.293·14-s + 0.258·15-s + 2.54·16-s − 0.425·17-s − 0.619·18-s − 1.98·19-s + 1.09·20-s − 0.0912·21-s − 0.528·22-s + 1.12·23-s − 1.55·24-s + 0.200·25-s − 0.513·26-s + 0.192·27-s − 0.386·28-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 + 2.62T + 2T^{2} \) |
| 7 | \( 1 + 0.417T + 7T^{2} \) |
| 11 | \( 1 - 0.944T + 11T^{2} \) |
| 13 | \( 1 - 0.996T + 13T^{2} \) |
| 17 | \( 1 + 1.75T + 17T^{2} \) |
| 19 | \( 1 + 8.64T + 19T^{2} \) |
| 23 | \( 1 - 5.41T + 23T^{2} \) |
| 29 | \( 1 - 0.444T + 29T^{2} \) |
| 31 | \( 1 + 2.08T + 31T^{2} \) |
| 37 | \( 1 - 2.35T + 37T^{2} \) |
| 41 | \( 1 + 1.86T + 41T^{2} \) |
| 43 | \( 1 + 10.9T + 43T^{2} \) |
| 47 | \( 1 + 6.04T + 47T^{2} \) |
| 53 | \( 1 - 9.40T + 53T^{2} \) |
| 59 | \( 1 - 0.577T + 59T^{2} \) |
| 61 | \( 1 + 2.21T + 61T^{2} \) |
| 67 | \( 1 - 16.3T + 67T^{2} \) |
| 71 | \( 1 - 1.90T + 71T^{2} \) |
| 73 | \( 1 + 14.1T + 73T^{2} \) |
| 79 | \( 1 + 6.00T + 79T^{2} \) |
| 83 | \( 1 - 5.94T + 83T^{2} \) |
| 89 | \( 1 - 2.36T + 89T^{2} \) |
| 97 | \( 1 - 4.24T + 97T^{2} \) |
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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
−8.137360442880722713007515327001, −7.03328206317463671200388953217, −6.71139200179121114695017905073, −6.06606587582490001222111774280, −4.87742449986503216567596316733, −3.73728127750813823877172582476, −2.76244039800255441367873962519, −2.05045676356524883785324260719, −1.30419441028632458888519535571, 0,
1.30419441028632458888519535571, 2.05045676356524883785324260719, 2.76244039800255441367873962519, 3.73728127750813823877172582476, 4.87742449986503216567596316733, 6.06606587582490001222111774280, 6.71139200179121114695017905073, 7.03328206317463671200388953217, 8.137360442880722713007515327001
