| L(s) = 1 | − 2.23·2-s + 3.23·3-s + 3.00·4-s − 1.61·5-s − 7.23·6-s − 0.381·7-s − 2.23·8-s + 7.47·9-s + 3.61·10-s + 9.70·12-s − 2.47·13-s + 0.854·14-s − 5.23·15-s − 0.999·16-s − 3.09·17-s − 16.7·18-s + 19-s − 4.85·20-s − 1.23·21-s − 0.618·23-s − 7.23·24-s − 2.38·25-s + 5.52·26-s + 14.4·27-s − 1.14·28-s + 8.47·29-s + 11.7·30-s + ⋯ |
| L(s) = 1 | − 1.58·2-s + 1.86·3-s + 1.50·4-s − 0.723·5-s − 2.95·6-s − 0.144·7-s − 0.790·8-s + 2.49·9-s + 1.14·10-s + 2.80·12-s − 0.685·13-s + 0.228·14-s − 1.35·15-s − 0.249·16-s − 0.749·17-s − 3.93·18-s + 0.229·19-s − 1.08·20-s − 0.269·21-s − 0.128·23-s − 1.47·24-s − 0.476·25-s + 1.08·26-s + 2.78·27-s − 0.216·28-s + 1.57·29-s + 2.13·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2299 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2299 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.396823146\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.396823146\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 2.23T + 2T^{2} \) |
| 3 | \( 1 - 3.23T + 3T^{2} \) |
| 5 | \( 1 + 1.61T + 5T^{2} \) |
| 7 | \( 1 + 0.381T + 7T^{2} \) |
| 13 | \( 1 + 2.47T + 13T^{2} \) |
| 17 | \( 1 + 3.09T + 17T^{2} \) |
| 23 | \( 1 + 0.618T + 23T^{2} \) |
| 29 | \( 1 - 8.47T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 11.7T + 41T^{2} \) |
| 43 | \( 1 + 3.09T + 43T^{2} \) |
| 47 | \( 1 - 8.61T + 47T^{2} \) |
| 53 | \( 1 - 1.52T + 53T^{2} \) |
| 59 | \( 1 - 12.4T + 59T^{2} \) |
| 61 | \( 1 - 8.61T + 61T^{2} \) |
| 67 | \( 1 + 8.47T + 67T^{2} \) |
| 71 | \( 1 - 11.7T + 71T^{2} \) |
| 73 | \( 1 - 3.52T + 73T^{2} \) |
| 79 | \( 1 - 2.94T + 79T^{2} \) |
| 83 | \( 1 - 5.32T + 83T^{2} \) |
| 89 | \( 1 + 3.23T + 89T^{2} \) |
| 97 | \( 1 + 12.1T + 97T^{2} \) |
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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
−8.822636460965089713695242898439, −8.340974137910220583871440336367, −7.84001890070502988200274477193, −7.19856762231216442533673179496, −6.54455841481763229551518168129, −4.66344281601966293857878682370, −3.91372492760317709257303183071, −2.75634239366939067647619850515, −2.21891341212330389881719538831, −0.893798454361588326556054343906,
0.893798454361588326556054343906, 2.21891341212330389881719538831, 2.75634239366939067647619850515, 3.91372492760317709257303183071, 4.66344281601966293857878682370, 6.54455841481763229551518168129, 7.19856762231216442533673179496, 7.84001890070502988200274477193, 8.340974137910220583871440336367, 8.822636460965089713695242898439
