L(s) = 1 | − 2.60·2-s + 4.76·4-s + 0.911·5-s − 3.17·7-s − 7.19·8-s − 2.37·10-s − 1.78·11-s + 0.673·13-s + 8.25·14-s + 9.17·16-s − 7.28·19-s + 4.34·20-s + 4.64·22-s + 2.71·23-s − 4.16·25-s − 1.75·26-s − 15.1·28-s + 9.27·29-s − 8.28·31-s − 9.47·32-s − 2.89·35-s − 5.17·37-s + 18.9·38-s − 6.55·40-s − 7.23·41-s + 11.5·43-s − 8.50·44-s + ⋯ |
L(s) = 1 | − 1.83·2-s + 2.38·4-s + 0.407·5-s − 1.20·7-s − 2.54·8-s − 0.750·10-s − 0.538·11-s + 0.186·13-s + 2.20·14-s + 2.29·16-s − 1.67·19-s + 0.971·20-s + 0.989·22-s + 0.566·23-s − 0.833·25-s − 0.343·26-s − 2.85·28-s + 1.72·29-s − 1.48·31-s − 1.67·32-s − 0.489·35-s − 0.851·37-s + 3.07·38-s − 1.03·40-s − 1.13·41-s + 1.75·43-s − 1.28·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7803 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7803 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3453654500\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3453654500\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 17 | \( 1 \) |
good | 2 | \( 1 + 2.60T + 2T^{2} \) |
| 5 | \( 1 - 0.911T + 5T^{2} \) |
| 7 | \( 1 + 3.17T + 7T^{2} \) |
| 11 | \( 1 + 1.78T + 11T^{2} \) |
| 13 | \( 1 - 0.673T + 13T^{2} \) |
| 19 | \( 1 + 7.28T + 19T^{2} \) |
| 23 | \( 1 - 2.71T + 23T^{2} \) |
| 29 | \( 1 - 9.27T + 29T^{2} \) |
| 31 | \( 1 + 8.28T + 31T^{2} \) |
| 37 | \( 1 + 5.17T + 37T^{2} \) |
| 41 | \( 1 + 7.23T + 41T^{2} \) |
| 43 | \( 1 - 11.5T + 43T^{2} \) |
| 47 | \( 1 + 3.74T + 47T^{2} \) |
| 53 | \( 1 - 0.748T + 53T^{2} \) |
| 59 | \( 1 - 1.82T + 59T^{2} \) |
| 61 | \( 1 - 5.50T + 61T^{2} \) |
| 67 | \( 1 + 2.09T + 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 + 0.334T + 73T^{2} \) |
| 79 | \( 1 + 11.5T + 79T^{2} \) |
| 83 | \( 1 + 17.2T + 83T^{2} \) |
| 89 | \( 1 + 8.11T + 89T^{2} \) |
| 97 | \( 1 - 2.10T + 97T^{2} \) |
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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
−8.118098223242144737222149572798, −7.15352448472115728529875090274, −6.73801210533561197056751586355, −6.14617044349827505524309186614, −5.42251575236893746967816792396, −4.11360132217722490151507502172, −3.08102744930432683895222382450, −2.41589639102977032181907864400, −1.60518911834065633228264126873, −0.37650357572927577012932589562,
0.37650357572927577012932589562, 1.60518911834065633228264126873, 2.41589639102977032181907864400, 3.08102744930432683895222382450, 4.11360132217722490151507502172, 5.42251575236893746967816792396, 6.14617044349827505524309186614, 6.73801210533561197056751586355, 7.15352448472115728529875090274, 8.118098223242144737222149572798