| 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
| \(L(1)\) |
\(\approx\) |
\(2.785029602\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.785029602\) |
| \(L(\frac{3}{2})\) |
|
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
−11.90996728307536461547427962426, −11.14173693563021234848340268957, −10.60391179595580500878044548809, −8.853845611970115535704271635628, −7.58088950642159042609611193781, −6.39611600354465424705721741657, −5.40278583616651330302747547773, −5.00838283148238606310744589930, −3.57316857708155970604272859617, −2.15038013449597672696834321091,
2.15038013449597672696834321091, 3.57316857708155970604272859617, 5.00838283148238606310744589930, 5.40278583616651330302747547773, 6.39611600354465424705721741657, 7.58088950642159042609611193781, 8.853845611970115535704271635628, 10.60391179595580500878044548809, 11.14173693563021234848340268957, 11.90996728307536461547427962426
