| L(s) = 1 | + 2.53·2-s − 3-s + 4.44·4-s + 0.994·5-s − 2.53·6-s + 4.88·7-s + 6.19·8-s + 9-s + 2.52·10-s + 11-s − 4.44·12-s + 4.29·13-s + 12.4·14-s − 0.994·15-s + 6.83·16-s − 3.82·17-s + 2.53·18-s − 2.93·19-s + 4.41·20-s − 4.88·21-s + 2.53·22-s − 5.41·23-s − 6.19·24-s − 4.01·25-s + 10.8·26-s − 27-s + 21.6·28-s + ⋯ |
| L(s) = 1 | + 1.79·2-s − 0.577·3-s + 2.22·4-s + 0.444·5-s − 1.03·6-s + 1.84·7-s + 2.18·8-s + 0.333·9-s + 0.797·10-s + 0.301·11-s − 1.28·12-s + 1.18·13-s + 3.31·14-s − 0.256·15-s + 1.70·16-s − 0.927·17-s + 0.598·18-s − 0.672·19-s + 0.986·20-s − 1.06·21-s + 0.541·22-s − 1.12·23-s − 1.26·24-s − 0.802·25-s + 2.13·26-s − 0.192·27-s + 4.10·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.831794057\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.831794057\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 - 2.53T + 2T^{2} \) |
| 5 | \( 1 - 0.994T + 5T^{2} \) |
| 7 | \( 1 - 4.88T + 7T^{2} \) |
| 13 | \( 1 - 4.29T + 13T^{2} \) |
| 17 | \( 1 + 3.82T + 17T^{2} \) |
| 19 | \( 1 + 2.93T + 19T^{2} \) |
| 23 | \( 1 + 5.41T + 23T^{2} \) |
| 29 | \( 1 + 2.31T + 29T^{2} \) |
| 31 | \( 1 + 6.92T + 31T^{2} \) |
| 37 | \( 1 - 0.949T + 37T^{2} \) |
| 41 | \( 1 + 4.66T + 41T^{2} \) |
| 43 | \( 1 - 4.23T + 43T^{2} \) |
| 47 | \( 1 - 8.27T + 47T^{2} \) |
| 53 | \( 1 + 11.5T + 53T^{2} \) |
| 59 | \( 1 - 5.12T + 59T^{2} \) |
| 67 | \( 1 + 6.83T + 67T^{2} \) |
| 71 | \( 1 - 4.72T + 71T^{2} \) |
| 73 | \( 1 - 15.9T + 73T^{2} \) |
| 79 | \( 1 + 6.64T + 79T^{2} \) |
| 83 | \( 1 + 8.78T + 83T^{2} \) |
| 89 | \( 1 - 0.0660T + 89T^{2} \) |
| 97 | \( 1 - 11.3T + 97T^{2} \) |
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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
−9.059342936803975410175664153792, −8.134797895911144960944890104946, −7.30418533255003908725724759559, −6.31723250562425365147329244361, −5.81238377162508449842808963390, −5.11248894247975601495683146511, −4.28360077809035585677451624077, −3.81874672078990584969602999722, −2.15352507900056904798411310996, −1.64493120072215073625106111653,
1.64493120072215073625106111653, 2.15352507900056904798411310996, 3.81874672078990584969602999722, 4.28360077809035585677451624077, 5.11248894247975601495683146511, 5.81238377162508449842808963390, 6.31723250562425365147329244361, 7.30418533255003908725724759559, 8.134797895911144960944890104946, 9.059342936803975410175664153792
