| L(s) = 1 | + 2.46·2-s − 2.90·3-s + 4.09·4-s − 0.0566·5-s − 7.17·6-s − 4.07·7-s + 5.17·8-s + 5.45·9-s − 0.139·10-s + 0.609·11-s − 11.9·12-s + 2.60·13-s − 10.0·14-s + 0.164·15-s + 4.59·16-s − 2.60·17-s + 13.4·18-s + 4.65·19-s − 0.232·20-s + 11.8·21-s + 1.50·22-s + 3.31·23-s − 15.0·24-s − 4.99·25-s + 6.44·26-s − 7.12·27-s − 16.6·28-s + ⋯ |
| L(s) = 1 | + 1.74·2-s − 1.67·3-s + 2.04·4-s − 0.0253·5-s − 2.93·6-s − 1.53·7-s + 1.83·8-s + 1.81·9-s − 0.0442·10-s + 0.183·11-s − 3.43·12-s + 0.723·13-s − 2.68·14-s + 0.0425·15-s + 1.14·16-s − 0.630·17-s + 3.17·18-s + 1.06·19-s − 0.0519·20-s + 2.58·21-s + 0.320·22-s + 0.691·23-s − 3.07·24-s − 0.999·25-s + 1.26·26-s − 1.37·27-s − 3.15·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.255332915\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.255332915\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 59 | \( 1 - T \) |
| good | 2 | \( 1 - 2.46T + 2T^{2} \) |
| 3 | \( 1 + 2.90T + 3T^{2} \) |
| 5 | \( 1 + 0.0566T + 5T^{2} \) |
| 7 | \( 1 + 4.07T + 7T^{2} \) |
| 11 | \( 1 - 0.609T + 11T^{2} \) |
| 13 | \( 1 - 2.60T + 13T^{2} \) |
| 17 | \( 1 + 2.60T + 17T^{2} \) |
| 19 | \( 1 - 4.65T + 19T^{2} \) |
| 23 | \( 1 - 3.31T + 23T^{2} \) |
| 29 | \( 1 + 1.69T + 29T^{2} \) |
| 31 | \( 1 - 3.94T + 31T^{2} \) |
| 37 | \( 1 - 3.50T + 37T^{2} \) |
| 41 | \( 1 + 10.4T + 41T^{2} \) |
| 43 | \( 1 + 4.70T + 43T^{2} \) |
| 47 | \( 1 + 8.93T + 47T^{2} \) |
| 53 | \( 1 + 0.577T + 53T^{2} \) |
| 61 | \( 1 - 13.2T + 61T^{2} \) |
| 67 | \( 1 - 2.09T + 67T^{2} \) |
| 71 | \( 1 + 4.81T + 71T^{2} \) |
| 73 | \( 1 - 11.2T + 73T^{2} \) |
| 79 | \( 1 - 7.50T + 79T^{2} \) |
| 83 | \( 1 - 13.0T + 83T^{2} \) |
| 89 | \( 1 + 7.01T + 89T^{2} \) |
| 97 | \( 1 + 2.40T + 97T^{2} \) |
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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
−15.36689762814051620251081564242, −13.55474392073197501679024315436, −12.95930860395454608230422153672, −11.92400272499216166055353452019, −11.23196723257498244534620311126, −9.894068549906690274672122077886, −6.81488755576059906160369376109, −6.21184657733260530828940271702, −5.11471649722704245802924848991, −3.58771324009033032494147405224,
3.58771324009033032494147405224, 5.11471649722704245802924848991, 6.21184657733260530828940271702, 6.81488755576059906160369376109, 9.894068549906690274672122077886, 11.23196723257498244534620311126, 11.92400272499216166055353452019, 12.95930860395454608230422153672, 13.55474392073197501679024315436, 15.36689762814051620251081564242
