| L(s) = 1 | − 0.612·2-s − 1.57·3-s − 1.62·4-s + 5-s + 0.964·6-s − 7-s + 2.22·8-s − 0.521·9-s − 0.612·10-s − 3.36·11-s + 2.55·12-s − 6.96·13-s + 0.612·14-s − 1.57·15-s + 1.88·16-s + 0.396·17-s + 0.319·18-s − 1.72·19-s − 1.62·20-s + 1.57·21-s + 2.06·22-s − 1.95·23-s − 3.49·24-s + 25-s + 4.26·26-s + 5.54·27-s + 1.62·28-s + ⋯ |
| L(s) = 1 | − 0.433·2-s − 0.909·3-s − 0.812·4-s + 0.447·5-s + 0.393·6-s − 0.377·7-s + 0.784·8-s − 0.173·9-s − 0.193·10-s − 1.01·11-s + 0.738·12-s − 1.93·13-s + 0.163·14-s − 0.406·15-s + 0.472·16-s + 0.0961·17-s + 0.0752·18-s − 0.396·19-s − 0.363·20-s + 0.343·21-s + 0.439·22-s − 0.407·23-s − 0.713·24-s + 0.200·25-s + 0.836·26-s + 1.06·27-s + 0.307·28-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 | \( 1 + 0.612T + 2T^{2} \) |
| 3 | \( 1 + 1.57T + 3T^{2} \) |
| 11 | \( 1 + 3.36T + 11T^{2} \) |
| 13 | \( 1 + 6.96T + 13T^{2} \) |
| 17 | \( 1 - 0.396T + 17T^{2} \) |
| 19 | \( 1 + 1.72T + 19T^{2} \) |
| 23 | \( 1 + 1.95T + 23T^{2} \) |
| 29 | \( 1 - 3.51T + 29T^{2} \) |
| 31 | \( 1 + 3.01T + 31T^{2} \) |
| 37 | \( 1 - 7.83T + 37T^{2} \) |
| 41 | \( 1 - 10.8T + 41T^{2} \) |
| 43 | \( 1 - 6.95T + 43T^{2} \) |
| 47 | \( 1 - 0.580T + 47T^{2} \) |
| 53 | \( 1 + 6.72T + 53T^{2} \) |
| 59 | \( 1 - 2.93T + 59T^{2} \) |
| 61 | \( 1 + 5.89T + 61T^{2} \) |
| 67 | \( 1 - 3.33T + 67T^{2} \) |
| 71 | \( 1 - 5.08T + 71T^{2} \) |
| 73 | \( 1 - 4.67T + 73T^{2} \) |
| 79 | \( 1 + 0.892T + 79T^{2} \) |
| 83 | \( 1 - 9.82T + 83T^{2} \) |
| 89 | \( 1 - 1.07T + 89T^{2} \) |
| 97 | \( 1 + 2.02T + 97T^{2} \) |
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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
−7.67653567928925490199937530405, −6.78568335135977953062940002461, −5.94272339783042905447341806997, −5.40029939881722323167386102118, −4.81377899104454483228913749660, −4.21470445657269146921813522149, −2.87879137587770753597229647692, −2.26342004729563146169796570356, −0.78066443572760645148484562980, 0,
0.78066443572760645148484562980, 2.26342004729563146169796570356, 2.87879137587770753597229647692, 4.21470445657269146921813522149, 4.81377899104454483228913749660, 5.40029939881722323167386102118, 5.94272339783042905447341806997, 6.78568335135977953062940002461, 7.67653567928925490199937530405
