| L(s) = 1 | − 2-s − 3-s − 4-s + 2·5-s + 6-s + 3·8-s + 9-s − 2·10-s + 11-s + 12-s + 13-s − 2·15-s − 16-s − 18-s − 4·19-s − 2·20-s − 22-s + 8·23-s − 3·24-s − 25-s − 26-s − 27-s + 10·29-s + 2·30-s − 5·32-s − 33-s − 36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.894·5-s + 0.408·6-s + 1.06·8-s + 1/3·9-s − 0.632·10-s + 0.301·11-s + 0.288·12-s + 0.277·13-s − 0.516·15-s − 1/4·16-s − 0.235·18-s − 0.917·19-s − 0.447·20-s − 0.213·22-s + 1.66·23-s − 0.612·24-s − 1/5·25-s − 0.196·26-s − 0.192·27-s + 1.85·29-s + 0.365·30-s − 0.883·32-s − 0.174·33-s − 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123981 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123981 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.77565256372549, −13.32978936602915, −12.86158420294654, −12.24946180774867, −11.97971857102897, −11.04394189560424, −10.75470589912075, −10.38844806473314, −9.927323147906790, −9.278801891734897, −9.013441792697107, −8.564832337374633, −7.931525235347072, −7.405754792941570, −6.669404382633610, −6.417031172977662, −5.818721907775743, −5.137295249454205, −4.686359225922958, −4.360992670544412, −3.394732487898351, −2.898962165428909, −1.848105346992005, −1.517995224277882, −0.8092826465064034, 0,
0.8092826465064034, 1.517995224277882, 1.848105346992005, 2.898962165428909, 3.394732487898351, 4.360992670544412, 4.686359225922958, 5.137295249454205, 5.818721907775743, 6.417031172977662, 6.669404382633610, 7.405754792941570, 7.931525235347072, 8.564832337374633, 9.013441792697107, 9.278801891734897, 9.927323147906790, 10.38844806473314, 10.75470589912075, 11.04394189560424, 11.97971857102897, 12.24946180774867, 12.86158420294654, 13.32978936602915, 13.77565256372549
