| L(s) = 1 | − 0.467·2-s − 3-s − 1.78·4-s − 0.0105·5-s + 0.467·6-s − 1.85·7-s + 1.76·8-s + 9-s + 0.00492·10-s − 1.01·11-s + 1.78·12-s + 2.69·13-s + 0.868·14-s + 0.0105·15-s + 2.73·16-s − 3.50·17-s − 0.467·18-s − 7.48·19-s + 0.0187·20-s + 1.85·21-s + 0.472·22-s + 23-s − 1.76·24-s − 4.99·25-s − 1.26·26-s − 27-s + 3.31·28-s + ⋯ |
| L(s) = 1 | − 0.330·2-s − 0.577·3-s − 0.890·4-s − 0.00471·5-s + 0.190·6-s − 0.702·7-s + 0.624·8-s + 0.333·9-s + 0.00155·10-s − 0.305·11-s + 0.514·12-s + 0.748·13-s + 0.232·14-s + 0.00272·15-s + 0.684·16-s − 0.851·17-s − 0.110·18-s − 1.71·19-s + 0.00420·20-s + 0.405·21-s + 0.100·22-s + 0.208·23-s − 0.360·24-s − 0.999·25-s − 0.247·26-s − 0.192·27-s + 0.626·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5535342018\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5535342018\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 0.467T + 2T^{2} \) |
| 5 | \( 1 + 0.0105T + 5T^{2} \) |
| 7 | \( 1 + 1.85T + 7T^{2} \) |
| 11 | \( 1 + 1.01T + 11T^{2} \) |
| 13 | \( 1 - 2.69T + 13T^{2} \) |
| 17 | \( 1 + 3.50T + 17T^{2} \) |
| 19 | \( 1 + 7.48T + 19T^{2} \) |
| 31 | \( 1 - 9.06T + 31T^{2} \) |
| 37 | \( 1 + 3.80T + 37T^{2} \) |
| 41 | \( 1 - 0.921T + 41T^{2} \) |
| 43 | \( 1 - 5.98T + 43T^{2} \) |
| 47 | \( 1 + 8.75T + 47T^{2} \) |
| 53 | \( 1 + 9.78T + 53T^{2} \) |
| 59 | \( 1 - 13.5T + 59T^{2} \) |
| 61 | \( 1 + 1.50T + 61T^{2} \) |
| 67 | \( 1 + 7.98T + 67T^{2} \) |
| 71 | \( 1 - 3.19T + 71T^{2} \) |
| 73 | \( 1 - 12.3T + 73T^{2} \) |
| 79 | \( 1 - 9.57T + 79T^{2} \) |
| 83 | \( 1 - 15.0T + 83T^{2} \) |
| 89 | \( 1 - 9.07T + 89T^{2} \) |
| 97 | \( 1 - 1.28T + 97T^{2} \) |
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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
−9.172508015321856862986396244489, −8.446198069199499584455106040226, −7.79833441804694260123478055298, −6.52842186671553463542159991577, −6.20302148909607423747866508118, −5.03963352856411221786734368494, −4.32404975511755867959041352947, −3.49533860763866633673915922970, −2.02880438009850536027589243947, −0.52773765035435024618070292896,
0.52773765035435024618070292896, 2.02880438009850536027589243947, 3.49533860763866633673915922970, 4.32404975511755867959041352947, 5.03963352856411221786734368494, 6.20302148909607423747866508118, 6.52842186671553463542159991577, 7.79833441804694260123478055298, 8.446198069199499584455106040226, 9.172508015321856862986396244489
