| L(s) = 1 | − 2.14i·2-s − i·3-s − 2.59·4-s + 1.55i·5-s − 2.14·6-s + 1.27i·8-s − 9-s + 3.34·10-s + (1.68 + 2.85i)11-s + 2.59i·12-s − 4.23·13-s + 1.55·15-s − 2.45·16-s − 0.829·17-s + 2.14i·18-s − 4.75·19-s + ⋯ |
| L(s) = 1 | − 1.51i·2-s − 0.577i·3-s − 1.29·4-s + 0.697i·5-s − 0.875·6-s + 0.451i·8-s − 0.333·9-s + 1.05·10-s + (0.509 + 0.860i)11-s + 0.749i·12-s − 1.17·13-s + 0.402·15-s − 0.613·16-s − 0.201·17-s + 0.505i·18-s − 1.09·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.112i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.112i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6935160748\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6935160748\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + iT \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (-1.68 - 2.85i)T \) |
| good | 2 | \( 1 + 2.14iT - 2T^{2} \) |
| 5 | \( 1 - 1.55iT - 5T^{2} \) |
| 13 | \( 1 + 4.23T + 13T^{2} \) |
| 17 | \( 1 + 0.829T + 17T^{2} \) |
| 19 | \( 1 + 4.75T + 19T^{2} \) |
| 23 | \( 1 + 1.60T + 23T^{2} \) |
| 29 | \( 1 + 0.853iT - 29T^{2} \) |
| 31 | \( 1 - 9.33iT - 31T^{2} \) |
| 37 | \( 1 - 3.41T + 37T^{2} \) |
| 41 | \( 1 + 0.308T + 41T^{2} \) |
| 43 | \( 1 - 6.72iT - 43T^{2} \) |
| 47 | \( 1 - 5.93iT - 47T^{2} \) |
| 53 | \( 1 + 4.16T + 53T^{2} \) |
| 59 | \( 1 - 8.37iT - 59T^{2} \) |
| 61 | \( 1 + 7.95T + 61T^{2} \) |
| 67 | \( 1 - 12.9T + 67T^{2} \) |
| 71 | \( 1 + 8.47T + 71T^{2} \) |
| 73 | \( 1 - 9.63T + 73T^{2} \) |
| 79 | \( 1 + 7.07iT - 79T^{2} \) |
| 83 | \( 1 + 1.82T + 83T^{2} \) |
| 89 | \( 1 + 1.92iT - 89T^{2} \) |
| 97 | \( 1 - 17.0iT - 97T^{2} \) |
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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
−9.616714990352543828105908616958, −8.947467824631510363613947910737, −7.84462197782962044520411752340, −6.92618691229675754448041421777, −6.42819647822145520387639886445, −4.91594577409126006144893896688, −4.17918290031283899386674536187, −2.99730364673687963083789111430, −2.36067658399273158475270017012, −1.41327858356430150027073874263,
0.26289881147401893063550263065, 2.34097521915073220259266749694, 3.88885028215883975493349433892, 4.65892675659924469626972352299, 5.36203201383122791897415524027, 6.14868976328070745569506306614, 6.87566826178533252521348208509, 7.88667090071476115021796169675, 8.461216651108408660263231578288, 9.136218952819471847980520317991
