L(s) = 1 | − 2.37·2-s + 1.45·3-s + 3.63·4-s − 5-s − 3.45·6-s + 1.94·7-s − 3.89·8-s − 0.887·9-s + 2.37·10-s − 6.31·11-s + 5.28·12-s − 2.12·13-s − 4.62·14-s − 1.45·15-s + 1.96·16-s − 0.676·17-s + 2.10·18-s + 6.14·19-s − 3.63·20-s + 2.82·21-s + 15.0·22-s + 1.31·23-s − 5.65·24-s + 25-s + 5.04·26-s − 5.65·27-s + 7.08·28-s + ⋯ |
L(s) = 1 | − 1.67·2-s + 0.839·3-s + 1.81·4-s − 0.447·5-s − 1.40·6-s + 0.735·7-s − 1.37·8-s − 0.295·9-s + 0.750·10-s − 1.90·11-s + 1.52·12-s − 0.589·13-s − 1.23·14-s − 0.375·15-s + 0.491·16-s − 0.164·17-s + 0.496·18-s + 1.41·19-s − 0.813·20-s + 0.617·21-s + 3.19·22-s + 0.274·23-s − 1.15·24-s + 0.200·25-s + 0.990·26-s − 1.08·27-s + 1.33·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7895026859\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7895026859\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + T \) |
| 241 | \( 1 - T \) |
good | 2 | \( 1 + 2.37T + 2T^{2} \) |
| 3 | \( 1 - 1.45T + 3T^{2} \) |
| 7 | \( 1 - 1.94T + 7T^{2} \) |
| 11 | \( 1 + 6.31T + 11T^{2} \) |
| 13 | \( 1 + 2.12T + 13T^{2} \) |
| 17 | \( 1 + 0.676T + 17T^{2} \) |
| 19 | \( 1 - 6.14T + 19T^{2} \) |
| 23 | \( 1 - 1.31T + 23T^{2} \) |
| 29 | \( 1 - 7.00T + 29T^{2} \) |
| 31 | \( 1 - 6.77T + 31T^{2} \) |
| 37 | \( 1 - 6.97T + 37T^{2} \) |
| 41 | \( 1 - 7.66T + 41T^{2} \) |
| 43 | \( 1 - 6.41T + 43T^{2} \) |
| 47 | \( 1 - 1.24T + 47T^{2} \) |
| 53 | \( 1 + 0.377T + 53T^{2} \) |
| 59 | \( 1 - 11.9T + 59T^{2} \) |
| 61 | \( 1 - 9.81T + 61T^{2} \) |
| 67 | \( 1 + 2.68T + 67T^{2} \) |
| 71 | \( 1 - 6.49T + 71T^{2} \) |
| 73 | \( 1 + 6.10T + 73T^{2} \) |
| 79 | \( 1 - 6.30T + 79T^{2} \) |
| 83 | \( 1 + 15.5T + 83T^{2} \) |
| 89 | \( 1 + 11.1T + 89T^{2} \) |
| 97 | \( 1 - 1.84T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.698616331554558424141769583804, −8.754775601709778823061929175443, −8.034516078014628225201927573125, −7.84568783975892052396953314787, −7.06653884237890313392205199659, −5.61261823931526669369100335750, −4.59719748243757393272997842871, −2.84668625824226097123153214232, −2.44532387587458856184262674782, −0.801144328405418277553896195892,
0.801144328405418277553896195892, 2.44532387587458856184262674782, 2.84668625824226097123153214232, 4.59719748243757393272997842871, 5.61261823931526669369100335750, 7.06653884237890313392205199659, 7.84568783975892052396953314787, 8.034516078014628225201927573125, 8.754775601709778823061929175443, 9.698616331554558424141769583804