L(s) = 1 | − 1.64·2-s + 1.19·3-s + 0.689·4-s − 5-s − 1.96·6-s − 2.90·7-s + 2.14·8-s − 1.56·9-s + 1.64·10-s + 11-s + 0.826·12-s − 4.40·13-s + 4.76·14-s − 1.19·15-s − 4.90·16-s − 3.56·17-s + 2.56·18-s − 3.72·19-s − 0.689·20-s − 3.48·21-s − 1.64·22-s − 6.43·23-s + 2.57·24-s + 25-s + 7.22·26-s − 5.46·27-s − 2.00·28-s + ⋯ |
L(s) = 1 | − 1.15·2-s + 0.691·3-s + 0.344·4-s − 0.447·5-s − 0.802·6-s − 1.09·7-s + 0.759·8-s − 0.521·9-s + 0.518·10-s + 0.301·11-s + 0.238·12-s − 1.22·13-s + 1.27·14-s − 0.309·15-s − 1.22·16-s − 0.864·17-s + 0.604·18-s − 0.855·19-s − 0.154·20-s − 0.760·21-s − 0.349·22-s − 1.34·23-s + 0.525·24-s + 0.200·25-s + 1.41·26-s − 1.05·27-s − 0.378·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2567359123\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2567359123\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 + T \) |
good | 2 | \( 1 + 1.64T + 2T^{2} \) |
| 3 | \( 1 - 1.19T + 3T^{2} \) |
| 7 | \( 1 + 2.90T + 7T^{2} \) |
| 13 | \( 1 + 4.40T + 13T^{2} \) |
| 17 | \( 1 + 3.56T + 17T^{2} \) |
| 19 | \( 1 + 3.72T + 19T^{2} \) |
| 23 | \( 1 + 6.43T + 23T^{2} \) |
| 29 | \( 1 + 7.11T + 29T^{2} \) |
| 31 | \( 1 + 6.72T + 31T^{2} \) |
| 37 | \( 1 - 8.74T + 37T^{2} \) |
| 41 | \( 1 - 7.21T + 41T^{2} \) |
| 43 | \( 1 - 12.0T + 43T^{2} \) |
| 47 | \( 1 + 0.194T + 47T^{2} \) |
| 53 | \( 1 + 0.243T + 53T^{2} \) |
| 59 | \( 1 + 3.10T + 59T^{2} \) |
| 61 | \( 1 - 11.3T + 61T^{2} \) |
| 67 | \( 1 + 4.95T + 67T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 79 | \( 1 + 5.81T + 79T^{2} \) |
| 83 | \( 1 - 11.3T + 83T^{2} \) |
| 89 | \( 1 + 2.86T + 89T^{2} \) |
| 97 | \( 1 - 2.45T + 97T^{2} \) |
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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
−8.616543219164350129806006554878, −7.67193935625476048032318690180, −7.48404520508315635974697107402, −6.46368387918017352975514797617, −5.67753870024639932716133129715, −4.34594463367373341137175580359, −3.86302857725025832460461113022, −2.64775524898317094320531870657, −2.04406382810389678485804204101, −0.31140976621060390796456196930,
0.31140976621060390796456196930, 2.04406382810389678485804204101, 2.64775524898317094320531870657, 3.86302857725025832460461113022, 4.34594463367373341137175580359, 5.67753870024639932716133129715, 6.46368387918017352975514797617, 7.48404520508315635974697107402, 7.67193935625476048032318690180, 8.616543219164350129806006554878