| L(s) = 1 | + 0.704·2-s − 3.11·3-s − 1.50·4-s − 2.19·6-s + 4.06·7-s − 2.46·8-s + 6.73·9-s − 0.231·11-s + 4.69·12-s − 2.35·13-s + 2.86·14-s + 1.27·16-s + 4.74·18-s + 7.38·19-s − 12.6·21-s − 0.162·22-s + 1.87·23-s + 7.69·24-s − 1.65·26-s − 11.6·27-s − 6.11·28-s + 4.04·29-s + 0.130·31-s + 5.82·32-s + 0.720·33-s − 10.1·36-s + 5.82·37-s + ⋯ |
| L(s) = 1 | + 0.497·2-s − 1.80·3-s − 0.752·4-s − 0.896·6-s + 1.53·7-s − 0.872·8-s + 2.24·9-s − 0.0696·11-s + 1.35·12-s − 0.653·13-s + 0.764·14-s + 0.317·16-s + 1.11·18-s + 1.69·19-s − 2.76·21-s − 0.0346·22-s + 0.391·23-s + 1.57·24-s − 0.325·26-s − 2.24·27-s − 1.15·28-s + 0.751·29-s + 0.0234·31-s + 1.03·32-s + 0.125·33-s − 1.68·36-s + 0.957·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.353660405\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.353660405\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 0.704T + 2T^{2} \) |
| 3 | \( 1 + 3.11T + 3T^{2} \) |
| 7 | \( 1 - 4.06T + 7T^{2} \) |
| 11 | \( 1 + 0.231T + 11T^{2} \) |
| 13 | \( 1 + 2.35T + 13T^{2} \) |
| 19 | \( 1 - 7.38T + 19T^{2} \) |
| 23 | \( 1 - 1.87T + 23T^{2} \) |
| 29 | \( 1 - 4.04T + 29T^{2} \) |
| 31 | \( 1 - 0.130T + 31T^{2} \) |
| 37 | \( 1 - 5.82T + 37T^{2} \) |
| 41 | \( 1 - 8.10T + 41T^{2} \) |
| 43 | \( 1 - 3.69T + 43T^{2} \) |
| 47 | \( 1 + 1.30T + 47T^{2} \) |
| 53 | \( 1 + 8.26T + 53T^{2} \) |
| 59 | \( 1 - 2.29T + 59T^{2} \) |
| 61 | \( 1 - 8.92T + 61T^{2} \) |
| 67 | \( 1 + 14.5T + 67T^{2} \) |
| 71 | \( 1 - 11.0T + 71T^{2} \) |
| 73 | \( 1 + 0.471T + 73T^{2} \) |
| 79 | \( 1 + 4.31T + 79T^{2} \) |
| 83 | \( 1 + 15.4T + 83T^{2} \) |
| 89 | \( 1 - 3.68T + 89T^{2} \) |
| 97 | \( 1 + 9.00T + 97T^{2} \) |
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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
−7.69098731328964734914046824872, −7.21801102245718344093269707982, −6.19393456331555870520595172277, −5.57358615600817643389366537238, −5.07530673726586719572733676757, −4.65738556102620911247539313562, −4.06344893037472523954427351760, −2.78793533728328553857377158847, −1.37411412236156984834886127148, −0.68897636282364820749264082328,
0.68897636282364820749264082328, 1.37411412236156984834886127148, 2.78793533728328553857377158847, 4.06344893037472523954427351760, 4.65738556102620911247539313562, 5.07530673726586719572733676757, 5.57358615600817643389366537238, 6.19393456331555870520595172277, 7.21801102245718344093269707982, 7.69098731328964734914046824872
