| L(s) = 1 | − 2.30·2-s − 0.474·3-s + 3.33·4-s + 1.09·6-s − 3.03·7-s − 3.07·8-s − 2.77·9-s + 2·11-s − 1.58·12-s + 1.42·13-s + 7.01·14-s + 0.441·16-s − 1.86·17-s + 6.40·18-s + 0.903·19-s + 1.44·21-s − 4.61·22-s + 3.32·23-s + 1.46·24-s − 3.29·26-s + 2.74·27-s − 10.1·28-s + 3.96·29-s − 6.43·31-s + 5.13·32-s − 0.949·33-s + 4.29·34-s + ⋯ |
| L(s) = 1 | − 1.63·2-s − 0.274·3-s + 1.66·4-s + 0.447·6-s − 1.14·7-s − 1.08·8-s − 0.924·9-s + 0.603·11-s − 0.456·12-s + 0.395·13-s + 1.87·14-s + 0.110·16-s − 0.451·17-s + 1.51·18-s + 0.207·19-s + 0.314·21-s − 0.984·22-s + 0.694·23-s + 0.298·24-s − 0.646·26-s + 0.527·27-s − 1.91·28-s + 0.735·29-s − 1.15·31-s + 0.907·32-s − 0.165·33-s + 0.736·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4377802329\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4377802329\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 + 2.30T + 2T^{2} \) |
| 3 | \( 1 + 0.474T + 3T^{2} \) |
| 7 | \( 1 + 3.03T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 1.42T + 13T^{2} \) |
| 17 | \( 1 + 1.86T + 17T^{2} \) |
| 19 | \( 1 - 0.903T + 19T^{2} \) |
| 23 | \( 1 - 3.32T + 23T^{2} \) |
| 29 | \( 1 - 3.96T + 29T^{2} \) |
| 31 | \( 1 + 6.43T + 31T^{2} \) |
| 37 | \( 1 + 3.82T + 37T^{2} \) |
| 41 | \( 1 + 1.83T + 41T^{2} \) |
| 43 | \( 1 - 3.59T + 43T^{2} \) |
| 47 | \( 1 - 4.79T + 47T^{2} \) |
| 53 | \( 1 - 9.50T + 53T^{2} \) |
| 59 | \( 1 - 10.6T + 59T^{2} \) |
| 61 | \( 1 - 14.2T + 61T^{2} \) |
| 67 | \( 1 - 10.6T + 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 + 0.267T + 73T^{2} \) |
| 79 | \( 1 + 8.57T + 79T^{2} \) |
| 83 | \( 1 - 12.6T + 83T^{2} \) |
| 89 | \( 1 + 4.76T + 89T^{2} \) |
| 97 | \( 1 - 9.95T + 97T^{2} \) |
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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
−10.45941843128280113859272478344, −9.577033963791122269248723893659, −8.937110227236741452254746723060, −8.323984522766985040189109951712, −7.02681633163367163597901498834, −6.55584052044442312452629707275, −5.45245414155533724379631942045, −3.67257849316276330073712996259, −2.42037034266320535485217120401, −0.71922159910025532786245851025,
0.71922159910025532786245851025, 2.42037034266320535485217120401, 3.67257849316276330073712996259, 5.45245414155533724379631942045, 6.55584052044442312452629707275, 7.02681633163367163597901498834, 8.323984522766985040189109951712, 8.937110227236741452254746723060, 9.577033963791122269248723893659, 10.45941843128280113859272478344
