L(s) = 1 | − 1.61·2-s + 3·3-s − 5.39·4-s − 4.25·5-s − 4.83·6-s − 1.08·7-s + 21.6·8-s + 9·9-s + 6.85·10-s + 22.8·11-s − 16.1·12-s − 68.3·13-s + 1.75·14-s − 12.7·15-s + 8.33·16-s + 31.2·17-s − 14.5·18-s − 35.7·19-s + 22.9·20-s − 3.26·21-s − 36.9·22-s + 23·23-s + 64.8·24-s − 106.·25-s + 110.·26-s + 27·27-s + 5.88·28-s + ⋯ |
L(s) = 1 | − 0.570·2-s + 0.577·3-s − 0.674·4-s − 0.380·5-s − 0.329·6-s − 0.0588·7-s + 0.955·8-s + 0.333·9-s + 0.216·10-s + 0.627·11-s − 0.389·12-s − 1.45·13-s + 0.0335·14-s − 0.219·15-s + 0.130·16-s + 0.445·17-s − 0.190·18-s − 0.431·19-s + 0.256·20-s − 0.0339·21-s − 0.357·22-s + 0.208·23-s + 0.551·24-s − 0.855·25-s + 0.832·26-s + 0.192·27-s + 0.0397·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.123995316\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.123995316\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 3T \) |
| 23 | \( 1 - 23T \) |
| 29 | \( 1 + 29T \) |
good | 2 | \( 1 + 1.61T + 8T^{2} \) |
| 5 | \( 1 + 4.25T + 125T^{2} \) |
| 7 | \( 1 + 1.08T + 343T^{2} \) |
| 11 | \( 1 - 22.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 68.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 31.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 35.7T + 6.85e3T^{2} \) |
| 31 | \( 1 - 222.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 215.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 241.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 151.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 573.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 272.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 136.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 522.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 102.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 883.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 685.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 374.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 338.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 547.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.76e3T + 9.12e5T^{2} \) |
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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
−8.887151041362061690907064551559, −8.027160059928657346594412550288, −7.55576686054736462900915448002, −6.72798602540817046145183065276, −5.49595253961110388038567550102, −4.54399732418569960781359181761, −3.95947993918837130723197838288, −2.85320557124606863783979356279, −1.70885361130221179181878089367, −0.52664042666231543221221520809,
0.52664042666231543221221520809, 1.70885361130221179181878089367, 2.85320557124606863783979356279, 3.95947993918837130723197838288, 4.54399732418569960781359181761, 5.49595253961110388038567550102, 6.72798602540817046145183065276, 7.55576686054736462900915448002, 8.027160059928657346594412550288, 8.887151041362061690907064551559