L(s) = 1 | − 0.0262·2-s + 3-s − 1.99·4-s − 2.77·5-s − 0.0262·6-s + 0.105·8-s + 9-s + 0.0729·10-s + 0.660·11-s − 1.99·12-s − 4.12·13-s − 2.77·15-s + 3.99·16-s − 4.12·17-s − 0.0262·18-s − 4.76·19-s + 5.55·20-s − 0.0173·22-s + 23-s + 0.105·24-s + 2.71·25-s + 0.108·26-s + 27-s − 1.78·29-s + 0.0729·30-s + 3.62·31-s − 0.314·32-s + ⋯ |
L(s) = 1 | − 0.0185·2-s + 0.577·3-s − 0.999·4-s − 1.24·5-s − 0.0107·6-s + 0.0371·8-s + 0.333·9-s + 0.0230·10-s + 0.199·11-s − 0.577·12-s − 1.14·13-s − 0.717·15-s + 0.998·16-s − 1.00·17-s − 0.00618·18-s − 1.09·19-s + 1.24·20-s − 0.00369·22-s + 0.208·23-s + 0.0214·24-s + 0.543·25-s + 0.0212·26-s + 0.192·27-s − 0.330·29-s + 0.0133·30-s + 0.650·31-s − 0.0556·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8070850470\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8070850470\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 2 | \( 1 + 0.0262T + 2T^{2} \) |
| 5 | \( 1 + 2.77T + 5T^{2} \) |
| 11 | \( 1 - 0.660T + 11T^{2} \) |
| 13 | \( 1 + 4.12T + 13T^{2} \) |
| 17 | \( 1 + 4.12T + 17T^{2} \) |
| 19 | \( 1 + 4.76T + 19T^{2} \) |
| 29 | \( 1 + 1.78T + 29T^{2} \) |
| 31 | \( 1 - 3.62T + 31T^{2} \) |
| 37 | \( 1 + 8.06T + 37T^{2} \) |
| 41 | \( 1 - 7.99T + 41T^{2} \) |
| 43 | \( 1 - 12.5T + 43T^{2} \) |
| 47 | \( 1 + 11.7T + 47T^{2} \) |
| 53 | \( 1 + 0.0608T + 53T^{2} \) |
| 59 | \( 1 - 8.98T + 59T^{2} \) |
| 61 | \( 1 + 2.71T + 61T^{2} \) |
| 67 | \( 1 + 2.68T + 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 - 14.2T + 73T^{2} \) |
| 79 | \( 1 + 6.94T + 79T^{2} \) |
| 83 | \( 1 + 9.40T + 83T^{2} \) |
| 89 | \( 1 - 5.92T + 89T^{2} \) |
| 97 | \( 1 + 2.53T + 97T^{2} \) |
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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
−8.530753345606571539091624714031, −8.033281969198478877434769529993, −7.32688954602656490616929566999, −6.57580485460605335953101837619, −5.34464603678961001105085251792, −4.39783715006137160936127148705, −4.17642078238059968923858944469, −3.19987258061096797714319587109, −2.13875619152704609144295115741, −0.50535875408761507247755398874,
0.50535875408761507247755398874, 2.13875619152704609144295115741, 3.19987258061096797714319587109, 4.17642078238059968923858944469, 4.39783715006137160936127148705, 5.34464603678961001105085251792, 6.57580485460605335953101837619, 7.32688954602656490616929566999, 8.033281969198478877434769529993, 8.530753345606571539091624714031