| L(s) = 1 | − 2.33·2-s − 3-s + 3.45·4-s + 4.00·5-s + 2.33·6-s − 7-s − 3.40·8-s + 9-s − 9.36·10-s + 2.66·11-s − 3.45·12-s − 4.55·13-s + 2.33·14-s − 4.00·15-s + 1.03·16-s + 4.20·17-s − 2.33·18-s − 4.30·19-s + 13.8·20-s + 21-s − 6.22·22-s + 5.00·23-s + 3.40·24-s + 11.0·25-s + 10.6·26-s − 27-s − 3.45·28-s + ⋯ |
| L(s) = 1 | − 1.65·2-s − 0.577·3-s + 1.72·4-s + 1.79·5-s + 0.953·6-s − 0.377·7-s − 1.20·8-s + 0.333·9-s − 2.96·10-s + 0.803·11-s − 0.997·12-s − 1.26·13-s + 0.624·14-s − 1.03·15-s + 0.259·16-s + 1.02·17-s − 0.550·18-s − 0.987·19-s + 3.09·20-s + 0.218·21-s − 1.32·22-s + 1.04·23-s + 0.694·24-s + 2.21·25-s + 2.08·26-s − 0.192·27-s − 0.653·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.067742714\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.067742714\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 - T \) |
| good | 2 | \( 1 + 2.33T + 2T^{2} \) |
| 5 | \( 1 - 4.00T + 5T^{2} \) |
| 11 | \( 1 - 2.66T + 11T^{2} \) |
| 13 | \( 1 + 4.55T + 13T^{2} \) |
| 17 | \( 1 - 4.20T + 17T^{2} \) |
| 19 | \( 1 + 4.30T + 19T^{2} \) |
| 23 | \( 1 - 5.00T + 23T^{2} \) |
| 29 | \( 1 - 4.42T + 29T^{2} \) |
| 31 | \( 1 - 0.736T + 31T^{2} \) |
| 37 | \( 1 - 5.24T + 37T^{2} \) |
| 41 | \( 1 - 4.50T + 41T^{2} \) |
| 43 | \( 1 + 3.93T + 43T^{2} \) |
| 47 | \( 1 + 6.53T + 47T^{2} \) |
| 53 | \( 1 - 7.91T + 53T^{2} \) |
| 59 | \( 1 - 2.01T + 59T^{2} \) |
| 61 | \( 1 + 10.8T + 61T^{2} \) |
| 67 | \( 1 + 2.01T + 67T^{2} \) |
| 71 | \( 1 + 9.71T + 71T^{2} \) |
| 73 | \( 1 - 3.46T + 73T^{2} \) |
| 79 | \( 1 - 0.606T + 79T^{2} \) |
| 83 | \( 1 - 4.98T + 83T^{2} \) |
| 89 | \( 1 - 3.16T + 89T^{2} \) |
| 97 | \( 1 - 16.2T + 97T^{2} \) |
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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
−7.83332345836962620808241285190, −7.14700024512932037168579061471, −6.45344020832349264596672040758, −6.18116438021937170393919117862, −5.24318043482451732290247974154, −4.53531022902041423830982177291, −3.00410487780599679222819020347, −2.27966662596828782587007703020, −1.49419429540664337789422759327, −0.72193750990986840022360560486,
0.72193750990986840022360560486, 1.49419429540664337789422759327, 2.27966662596828782587007703020, 3.00410487780599679222819020347, 4.53531022902041423830982177291, 5.24318043482451732290247974154, 6.18116438021937170393919117862, 6.45344020832349264596672040758, 7.14700024512932037168579061471, 7.83332345836962620808241285190
