L(s) = 1 | + 0.619·2-s − 1.61·4-s + 0.522·5-s − 1.16·7-s − 2.24·8-s + 0.324·10-s − 4.66·11-s − 1.29·13-s − 0.724·14-s + 1.84·16-s − 5.95·17-s + 7.77·19-s − 0.844·20-s − 2.88·22-s + 23-s − 4.72·25-s − 0.801·26-s + 1.88·28-s − 29-s − 5.43·31-s + 5.62·32-s − 3.68·34-s − 0.610·35-s + 7.45·37-s + 4.81·38-s − 1.17·40-s − 1.08·41-s + ⋯ |
L(s) = 1 | + 0.438·2-s − 0.807·4-s + 0.233·5-s − 0.441·7-s − 0.792·8-s + 0.102·10-s − 1.40·11-s − 0.358·13-s − 0.193·14-s + 0.460·16-s − 1.44·17-s + 1.78·19-s − 0.188·20-s − 0.616·22-s + 0.208·23-s − 0.945·25-s − 0.157·26-s + 0.356·28-s − 0.185·29-s − 0.976·31-s + 0.994·32-s − 0.632·34-s − 0.103·35-s + 1.22·37-s + 0.781·38-s − 0.185·40-s − 0.169·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9971177981\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9971177981\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
good | 2 | \( 1 - 0.619T + 2T^{2} \) |
| 5 | \( 1 - 0.522T + 5T^{2} \) |
| 7 | \( 1 + 1.16T + 7T^{2} \) |
| 11 | \( 1 + 4.66T + 11T^{2} \) |
| 13 | \( 1 + 1.29T + 13T^{2} \) |
| 17 | \( 1 + 5.95T + 17T^{2} \) |
| 19 | \( 1 - 7.77T + 19T^{2} \) |
| 31 | \( 1 + 5.43T + 31T^{2} \) |
| 37 | \( 1 - 7.45T + 37T^{2} \) |
| 41 | \( 1 + 1.08T + 41T^{2} \) |
| 43 | \( 1 - 5.14T + 43T^{2} \) |
| 47 | \( 1 + 8.76T + 47T^{2} \) |
| 53 | \( 1 + 7.42T + 53T^{2} \) |
| 59 | \( 1 + 6.63T + 59T^{2} \) |
| 61 | \( 1 - 1.32T + 61T^{2} \) |
| 67 | \( 1 + 12.1T + 67T^{2} \) |
| 71 | \( 1 - 2.41T + 71T^{2} \) |
| 73 | \( 1 - 3.43T + 73T^{2} \) |
| 79 | \( 1 - 16.4T + 79T^{2} \) |
| 83 | \( 1 - 14.8T + 83T^{2} \) |
| 89 | \( 1 - 15.4T + 89T^{2} \) |
| 97 | \( 1 - 3.33T + 97T^{2} \) |
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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
−7.82210855490935644029813705245, −7.64378616109796097317566818171, −6.45346239784255870037052740573, −5.84216479638456041193306906441, −5.03876770771624903362009874718, −4.69426798722769263162201392708, −3.57886116390309657464083642246, −2.97737302494367395730014924760, −2.02872666164600751805359395065, −0.46979973477744078165135996512,
0.46979973477744078165135996512, 2.02872666164600751805359395065, 2.97737302494367395730014924760, 3.57886116390309657464083642246, 4.69426798722769263162201392708, 5.03876770771624903362009874718, 5.84216479638456041193306906441, 6.45346239784255870037052740573, 7.64378616109796097317566818171, 7.82210855490935644029813705245