| L(s) = 1 | + 1.61·2-s + 0.381·3-s + 0.618·4-s + 3.23·5-s + 0.618·6-s + 3·7-s − 2.23·8-s − 2.85·9-s + 5.23·10-s − 1.61·11-s + 0.236·12-s − 13-s + 4.85·14-s + 1.23·15-s − 4.85·16-s + 0.763·17-s − 4.61·18-s + 2.00·20-s + 1.14·21-s − 2.61·22-s + 5.38·23-s − 0.854·24-s + 5.47·25-s − 1.61·26-s − 2.23·27-s + 1.85·28-s − 3.61·29-s + ⋯ |
| L(s) = 1 | + 1.14·2-s + 0.220·3-s + 0.309·4-s + 1.44·5-s + 0.252·6-s + 1.13·7-s − 0.790·8-s − 0.951·9-s + 1.65·10-s − 0.487·11-s + 0.0681·12-s − 0.277·13-s + 1.29·14-s + 0.319·15-s − 1.21·16-s + 0.185·17-s − 1.08·18-s + 0.447·20-s + 0.250·21-s − 0.558·22-s + 1.12·23-s − 0.174·24-s + 1.09·25-s − 0.317·26-s − 0.430·27-s + 0.350·28-s − 0.671·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.771031028\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.771031028\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 \) |
| good | 2 | \( 1 - 1.61T + 2T^{2} \) |
| 3 | \( 1 - 0.381T + 3T^{2} \) |
| 5 | \( 1 - 3.23T + 5T^{2} \) |
| 7 | \( 1 - 3T + 7T^{2} \) |
| 11 | \( 1 + 1.61T + 11T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 - 0.763T + 17T^{2} \) |
| 23 | \( 1 - 5.38T + 23T^{2} \) |
| 29 | \( 1 + 3.61T + 29T^{2} \) |
| 31 | \( 1 + 8.85T + 31T^{2} \) |
| 37 | \( 1 - 8.85T + 37T^{2} \) |
| 41 | \( 1 + 3T + 41T^{2} \) |
| 43 | \( 1 + 0.145T + 43T^{2} \) |
| 47 | \( 1 - 3T + 47T^{2} \) |
| 53 | \( 1 + 6.32T + 53T^{2} \) |
| 59 | \( 1 - 0.326T + 59T^{2} \) |
| 61 | \( 1 + 10.2T + 61T^{2} \) |
| 67 | \( 1 + 7T + 67T^{2} \) |
| 71 | \( 1 + 7.47T + 71T^{2} \) |
| 73 | \( 1 + 2.70T + 73T^{2} \) |
| 79 | \( 1 - 13.4T + 79T^{2} \) |
| 83 | \( 1 - 8.47T + 83T^{2} \) |
| 89 | \( 1 + 7.76T + 89T^{2} \) |
| 97 | \( 1 - 13.8T + 97T^{2} \) |
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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
−11.49788157458842022587931033710, −10.77748306074970311755371727392, −9.454831385739857431043096854179, −8.829671547173231451633876685519, −7.59258639385275656373452645746, −6.11007784620556506954285677917, −5.44086334218636031304886039989, −4.74930228065550701771302606444, −3.13206776420011051494926956226, −2.04548392419334581624899500641,
2.04548392419334581624899500641, 3.13206776420011051494926956226, 4.74930228065550701771302606444, 5.44086334218636031304886039989, 6.11007784620556506954285677917, 7.59258639385275656373452645746, 8.829671547173231451633876685519, 9.454831385739857431043096854179, 10.77748306074970311755371727392, 11.49788157458842022587931033710
