| L(s) = 1 | − 2.51·2-s − 3-s + 4.30·4-s + 2.07·5-s + 2.51·6-s − 0.329·7-s − 5.79·8-s + 9-s − 5.20·10-s + 4.89·11-s − 4.30·12-s + 2.21·13-s + 0.827·14-s − 2.07·15-s + 5.93·16-s + 4.14·17-s − 2.51·18-s + 0.434·19-s + 8.93·20-s + 0.329·21-s − 12.2·22-s + 23-s + 5.79·24-s − 0.696·25-s − 5.56·26-s − 27-s − 1.41·28-s + ⋯ |
| L(s) = 1 | − 1.77·2-s − 0.577·3-s + 2.15·4-s + 0.927·5-s + 1.02·6-s − 0.124·7-s − 2.04·8-s + 0.333·9-s − 1.64·10-s + 1.47·11-s − 1.24·12-s + 0.614·13-s + 0.221·14-s − 0.535·15-s + 1.48·16-s + 1.00·17-s − 0.591·18-s + 0.0997·19-s + 1.99·20-s + 0.0718·21-s − 2.62·22-s + 0.208·23-s + 1.18·24-s − 0.139·25-s − 1.09·26-s − 0.192·27-s − 0.268·28-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
| \(L(1)\) |
\(\approx\) |
\(0.9157669624\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9157669624\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.51T + 2T^{2} \) |
| 5 | \( 1 - 2.07T + 5T^{2} \) |
| 7 | \( 1 + 0.329T + 7T^{2} \) |
| 11 | \( 1 - 4.89T + 11T^{2} \) |
| 13 | \( 1 - 2.21T + 13T^{2} \) |
| 17 | \( 1 - 4.14T + 17T^{2} \) |
| 19 | \( 1 - 0.434T + 19T^{2} \) |
| 31 | \( 1 - 11.0T + 31T^{2} \) |
| 37 | \( 1 - 0.655T + 37T^{2} \) |
| 41 | \( 1 - 7.37T + 41T^{2} \) |
| 43 | \( 1 - 0.463T + 43T^{2} \) |
| 47 | \( 1 - 4.14T + 47T^{2} \) |
| 53 | \( 1 - 2.30T + 53T^{2} \) |
| 59 | \( 1 - 9.22T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 + 14.6T + 67T^{2} \) |
| 71 | \( 1 - 12.1T + 71T^{2} \) |
| 73 | \( 1 - 7.93T + 73T^{2} \) |
| 79 | \( 1 + 6.32T + 79T^{2} \) |
| 83 | \( 1 + 16.4T + 83T^{2} \) |
| 89 | \( 1 - 7.88T + 89T^{2} \) |
| 97 | \( 1 + 15.8T + 97T^{2} \) |
| 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
−9.391420514486406058610907740102, −8.542231752885412374541509984816, −7.75724794841458147856908706523, −6.80165756669254792788198385414, −6.28161892449500041704157172921, −5.64147875691332257730429629635, −4.18649873379842852952066763340, −2.84450399712221432342739896905, −1.58158014552732901277358062857, −0.960119216840770139771604636380,
0.960119216840770139771604636380, 1.58158014552732901277358062857, 2.84450399712221432342739896905, 4.18649873379842852952066763340, 5.64147875691332257730429629635, 6.28161892449500041704157172921, 6.80165756669254792788198385414, 7.75724794841458147856908706523, 8.542231752885412374541509984816, 9.391420514486406058610907740102
