| L(s) = 1 | − 2.51·2-s + 1.29·3-s + 4.30·4-s − 2.92·5-s − 3.26·6-s + 2.29·7-s − 5.77·8-s − 1.31·9-s + 7.33·10-s − 1.13·11-s + 5.58·12-s − 0.0412·13-s − 5.75·14-s − 3.79·15-s + 5.89·16-s + 4.38·17-s + 3.29·18-s + 1.65·19-s − 12.5·20-s + 2.97·21-s + 2.84·22-s − 8.34·23-s − 7.49·24-s + 3.53·25-s + 0.103·26-s − 5.60·27-s + 9.85·28-s + ⋯ |
| L(s) = 1 | − 1.77·2-s + 0.750·3-s + 2.15·4-s − 1.30·5-s − 1.33·6-s + 0.865·7-s − 2.04·8-s − 0.437·9-s + 2.31·10-s − 0.341·11-s + 1.61·12-s − 0.0114·13-s − 1.53·14-s − 0.979·15-s + 1.47·16-s + 1.06·17-s + 0.776·18-s + 0.380·19-s − 2.80·20-s + 0.649·21-s + 0.605·22-s − 1.74·23-s − 1.53·24-s + 0.706·25-s + 0.0202·26-s − 1.07·27-s + 1.86·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 31 | \( 1 \) |
| good | 2 | \( 1 + 2.51T + 2T^{2} \) |
| 3 | \( 1 - 1.29T + 3T^{2} \) |
| 5 | \( 1 + 2.92T + 5T^{2} \) |
| 7 | \( 1 - 2.29T + 7T^{2} \) |
| 11 | \( 1 + 1.13T + 11T^{2} \) |
| 13 | \( 1 + 0.0412T + 13T^{2} \) |
| 17 | \( 1 - 4.38T + 17T^{2} \) |
| 19 | \( 1 - 1.65T + 19T^{2} \) |
| 23 | \( 1 + 8.34T + 23T^{2} \) |
| 29 | \( 1 - 4.48T + 29T^{2} \) |
| 37 | \( 1 + 5.96T + 37T^{2} \) |
| 41 | \( 1 + 7.20T + 41T^{2} \) |
| 43 | \( 1 + 3.63T + 43T^{2} \) |
| 47 | \( 1 - 4.21T + 47T^{2} \) |
| 53 | \( 1 + 9.97T + 53T^{2} \) |
| 59 | \( 1 - 1.49T + 59T^{2} \) |
| 61 | \( 1 - 6.42T + 61T^{2} \) |
| 67 | \( 1 - 1.87T + 67T^{2} \) |
| 71 | \( 1 + 6.10T + 71T^{2} \) |
| 73 | \( 1 - 14.3T + 73T^{2} \) |
| 79 | \( 1 + 14.8T + 79T^{2} \) |
| 83 | \( 1 - 10.0T + 83T^{2} \) |
| 89 | \( 1 - 0.172T + 89T^{2} \) |
| 97 | \( 1 + 11.6T + 97T^{2} \) |
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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
−9.484587354223820246811104510297, −8.430630073959601296517883258627, −8.065583960369565884138761426582, −7.78634085206227338576453596247, −6.73955729138696301399905186999, −5.34973544657860862781319149895, −3.86885144246652684675176141815, −2.82939762479986331261199968439, −1.59329056221677408367523576292, 0,
1.59329056221677408367523576292, 2.82939762479986331261199968439, 3.86885144246652684675176141815, 5.34973544657860862781319149895, 6.73955729138696301399905186999, 7.78634085206227338576453596247, 8.065583960369565884138761426582, 8.430630073959601296517883258627, 9.484587354223820246811104510297
