L(s) = 1 | + 1.00·2-s − 1.61·3-s − 0.983·4-s + 5-s − 1.63·6-s + 7-s − 3.00·8-s − 0.379·9-s + 1.00·10-s − 4.04·11-s + 1.59·12-s − 1.03·13-s + 1.00·14-s − 1.61·15-s − 1.06·16-s − 0.400·17-s − 0.382·18-s − 0.496·19-s − 0.983·20-s − 1.61·21-s − 4.07·22-s + 5.90·23-s + 4.86·24-s + 25-s − 1.03·26-s + 5.47·27-s − 0.983·28-s + ⋯ |
L(s) = 1 | + 0.713·2-s − 0.934·3-s − 0.491·4-s + 0.447·5-s − 0.666·6-s + 0.377·7-s − 1.06·8-s − 0.126·9-s + 0.318·10-s − 1.21·11-s + 0.459·12-s − 0.285·13-s + 0.269·14-s − 0.417·15-s − 0.266·16-s − 0.0972·17-s − 0.0902·18-s − 0.113·19-s − 0.219·20-s − 0.353·21-s − 0.868·22-s + 1.23·23-s + 0.994·24-s + 0.200·25-s − 0.203·26-s + 1.05·27-s − 0.185·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 229 | \( 1 + T \) |
good | 2 | \( 1 - 1.00T + 2T^{2} \) |
| 3 | \( 1 + 1.61T + 3T^{2} \) |
| 11 | \( 1 + 4.04T + 11T^{2} \) |
| 13 | \( 1 + 1.03T + 13T^{2} \) |
| 17 | \( 1 + 0.400T + 17T^{2} \) |
| 19 | \( 1 + 0.496T + 19T^{2} \) |
| 23 | \( 1 - 5.90T + 23T^{2} \) |
| 29 | \( 1 + 3.94T + 29T^{2} \) |
| 31 | \( 1 - 7.68T + 31T^{2} \) |
| 37 | \( 1 - 10.0T + 37T^{2} \) |
| 41 | \( 1 - 4.56T + 41T^{2} \) |
| 43 | \( 1 - 6.96T + 43T^{2} \) |
| 47 | \( 1 + 7.20T + 47T^{2} \) |
| 53 | \( 1 + 5.48T + 53T^{2} \) |
| 59 | \( 1 + 12.9T + 59T^{2} \) |
| 61 | \( 1 - 3.68T + 61T^{2} \) |
| 67 | \( 1 + 12.4T + 67T^{2} \) |
| 71 | \( 1 - 11.8T + 71T^{2} \) |
| 73 | \( 1 + 9.91T + 73T^{2} \) |
| 79 | \( 1 + 3.76T + 79T^{2} \) |
| 83 | \( 1 - 3.48T + 83T^{2} \) |
| 89 | \( 1 - 11.8T + 89T^{2} \) |
| 97 | \( 1 + 17.0T + 97T^{2} \) |
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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
−7.48260397815280103455879222205, −6.33269144506746260346251861099, −6.03700107128424206677096921089, −5.20372172447162899720328130262, −4.90704213933219523229875529015, −4.25660700021896039731897289384, −2.99961658033716002974446869582, −2.55816467587028229698376962066, −1.05128661818663000051710967037, 0,
1.05128661818663000051710967037, 2.55816467587028229698376962066, 2.99961658033716002974446869582, 4.25660700021896039731897289384, 4.90704213933219523229875529015, 5.20372172447162899720328130262, 6.03700107128424206677096921089, 6.33269144506746260346251861099, 7.48260397815280103455879222205