| L(s) = 1 | + 2.41·2-s + 1.08·3-s + 3.82·4-s − 0.765·5-s + 2.61·6-s − 2.61·7-s + 4.41·8-s − 1.82·9-s − 1.84·10-s + 2.61·11-s + 4.14·12-s + 1.41·13-s − 6.30·14-s − 0.828·15-s + 2.99·16-s − 4.41·18-s − 0.828·19-s − 2.93·20-s − 2.82·21-s + 6.30·22-s − 4.77·23-s + 4.77·24-s − 4.41·25-s + 3.41·26-s − 5.22·27-s − 10.0·28-s + 0.317·29-s + ⋯ |
| L(s) = 1 | + 1.70·2-s + 0.624·3-s + 1.91·4-s − 0.342·5-s + 1.06·6-s − 0.987·7-s + 1.56·8-s − 0.609·9-s − 0.584·10-s + 0.787·11-s + 1.19·12-s + 0.392·13-s − 1.68·14-s − 0.213·15-s + 0.749·16-s − 1.04·18-s − 0.190·19-s − 0.655·20-s − 0.617·21-s + 1.34·22-s − 0.996·23-s + 0.975·24-s − 0.882·25-s + 0.669·26-s − 1.00·27-s − 1.89·28-s + 0.0588·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.246049229\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.246049229\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 - 2.41T + 2T^{2} \) |
| 3 | \( 1 - 1.08T + 3T^{2} \) |
| 5 | \( 1 + 0.765T + 5T^{2} \) |
| 7 | \( 1 + 2.61T + 7T^{2} \) |
| 11 | \( 1 - 2.61T + 11T^{2} \) |
| 13 | \( 1 - 1.41T + 13T^{2} \) |
| 19 | \( 1 + 0.828T + 19T^{2} \) |
| 23 | \( 1 + 4.77T + 23T^{2} \) |
| 29 | \( 1 - 0.317T + 29T^{2} \) |
| 31 | \( 1 - 7.83T + 31T^{2} \) |
| 37 | \( 1 - 9.23T + 37T^{2} \) |
| 41 | \( 1 - 1.21T + 41T^{2} \) |
| 43 | \( 1 - 0.828T + 43T^{2} \) |
| 47 | \( 1 - 5.17T + 47T^{2} \) |
| 53 | \( 1 - 1.41T + 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 - 3.82T + 61T^{2} \) |
| 67 | \( 1 + 1.17T + 67T^{2} \) |
| 71 | \( 1 + 5.41T + 71T^{2} \) |
| 73 | \( 1 + 12.9T + 73T^{2} \) |
| 79 | \( 1 + 4.77T + 79T^{2} \) |
| 83 | \( 1 - 11.6T + 83T^{2} \) |
| 89 | \( 1 - 6.58T + 89T^{2} \) |
| 97 | \( 1 + 10.3T + 97T^{2} \) |
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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
−11.92664504259331543469692957157, −11.44490264332367850376922939218, −10.03257715625309243538036058376, −8.908298033133909763344515365998, −7.71911200805315923193127440952, −6.41956642714792924678211150818, −5.87746849482190484082708609550, −4.28104241987363091501430920742, −3.53554923684938663345719112771, −2.50735046904301974326543780905,
2.50735046904301974326543780905, 3.53554923684938663345719112771, 4.28104241987363091501430920742, 5.87746849482190484082708609550, 6.41956642714792924678211150818, 7.71911200805315923193127440952, 8.908298033133909763344515365998, 10.03257715625309243538036058376, 11.44490264332367850376922939218, 11.92664504259331543469692957157
