L(s) = 1 | − 2-s − 3-s + 4-s − 3.68·5-s + 6-s + 5.03·7-s − 8-s + 9-s + 3.68·10-s − 6.07·11-s − 12-s + 13-s − 5.03·14-s + 3.68·15-s + 16-s + 0.687·17-s − 18-s + 4.84·19-s − 3.68·20-s − 5.03·21-s + 6.07·22-s + 5.54·23-s + 24-s + 8.56·25-s − 26-s − 27-s + 5.03·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.64·5-s + 0.408·6-s + 1.90·7-s − 0.353·8-s + 0.333·9-s + 1.16·10-s − 1.83·11-s − 0.288·12-s + 0.277·13-s − 1.34·14-s + 0.951·15-s + 0.250·16-s + 0.166·17-s − 0.235·18-s + 1.11·19-s − 0.823·20-s − 1.09·21-s + 1.29·22-s + 1.15·23-s + 0.204·24-s + 1.71·25-s − 0.196·26-s − 0.192·27-s + 0.951·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8714678524\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8714678524\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
good | 5 | \( 1 + 3.68T + 5T^{2} \) |
| 7 | \( 1 - 5.03T + 7T^{2} \) |
| 11 | \( 1 + 6.07T + 11T^{2} \) |
| 17 | \( 1 - 0.687T + 17T^{2} \) |
| 19 | \( 1 - 4.84T + 19T^{2} \) |
| 23 | \( 1 - 5.54T + 23T^{2} \) |
| 29 | \( 1 - 6.66T + 29T^{2} \) |
| 31 | \( 1 + 3.81T + 31T^{2} \) |
| 37 | \( 1 - 0.460T + 37T^{2} \) |
| 41 | \( 1 + 7.54T + 41T^{2} \) |
| 43 | \( 1 + 8.38T + 43T^{2} \) |
| 47 | \( 1 - 5.56T + 47T^{2} \) |
| 53 | \( 1 - 7.78T + 53T^{2} \) |
| 59 | \( 1 + 5.42T + 59T^{2} \) |
| 61 | \( 1 + 6.72T + 61T^{2} \) |
| 67 | \( 1 - 10.8T + 67T^{2} \) |
| 71 | \( 1 - 14.1T + 71T^{2} \) |
| 73 | \( 1 + 11.0T + 73T^{2} \) |
| 79 | \( 1 - 1.41T + 79T^{2} \) |
| 83 | \( 1 + 5.18T + 83T^{2} \) |
| 89 | \( 1 - 4.44T + 89T^{2} \) |
| 97 | \( 1 + 14.0T + 97T^{2} \) |
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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
−7.81737255574879389665798380119, −7.44444687805686712137990452939, −6.83528347814462368446169044568, −5.42435508208345545234603631589, −5.11736598924546977258185594185, −4.49653981718683555054004771507, −3.46744431866091766035131811315, −2.61456681695898550338460482247, −1.41253399391008934625001735483, −0.57679898831678751243497204913,
0.57679898831678751243497204913, 1.41253399391008934625001735483, 2.61456681695898550338460482247, 3.46744431866091766035131811315, 4.49653981718683555054004771507, 5.11736598924546977258185594185, 5.42435508208345545234603631589, 6.83528347814462368446169044568, 7.44444687805686712137990452939, 7.81737255574879389665798380119