| L(s) = 1 | + 2.49·2-s − 2.47·3-s + 4.23·4-s − 1.14·5-s − 6.17·6-s + 0.265·7-s + 5.56·8-s + 3.11·9-s − 2.86·10-s − 0.291·11-s − 10.4·12-s + 4.52·13-s + 0.661·14-s + 2.84·15-s + 5.44·16-s + 5.49·17-s + 7.77·18-s + 6.95·19-s − 4.86·20-s − 0.655·21-s − 0.727·22-s + 2.96·23-s − 13.7·24-s − 3.67·25-s + 11.2·26-s − 0.286·27-s + 1.12·28-s + ⋯ |
| L(s) = 1 | + 1.76·2-s − 1.42·3-s + 2.11·4-s − 0.513·5-s − 2.52·6-s + 0.100·7-s + 1.96·8-s + 1.03·9-s − 0.907·10-s − 0.0879·11-s − 3.02·12-s + 1.25·13-s + 0.176·14-s + 0.733·15-s + 1.36·16-s + 1.33·17-s + 1.83·18-s + 1.59·19-s − 1.08·20-s − 0.143·21-s − 0.155·22-s + 0.618·23-s − 2.81·24-s − 0.735·25-s + 2.21·26-s − 0.0552·27-s + 0.211·28-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
| \(L(1)\) |
\(\approx\) |
\(2.881440565\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.881440565\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.49T + 2T^{2} \) |
| 3 | \( 1 + 2.47T + 3T^{2} \) |
| 5 | \( 1 + 1.14T + 5T^{2} \) |
| 7 | \( 1 - 0.265T + 7T^{2} \) |
| 11 | \( 1 + 0.291T + 11T^{2} \) |
| 13 | \( 1 - 4.52T + 13T^{2} \) |
| 17 | \( 1 - 5.49T + 17T^{2} \) |
| 19 | \( 1 - 6.95T + 19T^{2} \) |
| 23 | \( 1 - 2.96T + 23T^{2} \) |
| 29 | \( 1 + 3.32T + 29T^{2} \) |
| 37 | \( 1 + 4.07T + 37T^{2} \) |
| 41 | \( 1 - 7.49T + 41T^{2} \) |
| 43 | \( 1 - 8.05T + 43T^{2} \) |
| 47 | \( 1 - 5.51T + 47T^{2} \) |
| 53 | \( 1 + 4.75T + 53T^{2} \) |
| 59 | \( 1 + 2.35T + 59T^{2} \) |
| 61 | \( 1 + 10.0T + 61T^{2} \) |
| 67 | \( 1 + 3.58T + 67T^{2} \) |
| 71 | \( 1 - 10.8T + 71T^{2} \) |
| 73 | \( 1 + 2.27T + 73T^{2} \) |
| 79 | \( 1 + 3.79T + 79T^{2} \) |
| 83 | \( 1 - 2.79T + 83T^{2} \) |
| 89 | \( 1 + 3.20T + 89T^{2} \) |
| 97 | \( 1 - 2.01T + 97T^{2} \) |
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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
−10.71266341049430674537478092850, −9.428833947251081132681148322140, −7.83244653316419576330617568411, −7.14490477213935664124002624052, −6.04396100125625085608745799301, −5.67179948428029802495443929114, −4.91735484067636694032458029244, −3.91267720446921018587512311540, −3.13013962216912147376389114347, −1.22757920293218799713496602599,
1.22757920293218799713496602599, 3.13013962216912147376389114347, 3.91267720446921018587512311540, 4.91735484067636694032458029244, 5.67179948428029802495443929114, 6.04396100125625085608745799301, 7.14490477213935664124002624052, 7.83244653316419576330617568411, 9.428833947251081132681148322140, 10.71266341049430674537478092850
