| L(s) = 1 | − 2.12·2-s − 0.167·3-s + 2.52·4-s − 1.92·5-s + 0.356·6-s + 0.0410·7-s − 1.11·8-s − 2.97·9-s + 4.09·10-s + 3.07·11-s − 0.422·12-s − 13-s − 0.0873·14-s + 0.322·15-s − 2.68·16-s − 4.74·17-s + 6.31·18-s − 1.80·19-s − 4.85·20-s − 0.00688·21-s − 6.54·22-s − 4.58·23-s + 0.186·24-s − 1.29·25-s + 2.12·26-s + 1.00·27-s + 0.103·28-s + ⋯ |
| L(s) = 1 | − 1.50·2-s − 0.0967·3-s + 1.26·4-s − 0.861·5-s + 0.145·6-s + 0.0155·7-s − 0.392·8-s − 0.990·9-s + 1.29·10-s + 0.927·11-s − 0.122·12-s − 0.277·13-s − 0.0233·14-s + 0.0833·15-s − 0.670·16-s − 1.14·17-s + 1.48·18-s − 0.413·19-s − 1.08·20-s − 0.00150·21-s − 1.39·22-s − 0.955·23-s + 0.0379·24-s − 0.258·25-s + 0.417·26-s + 0.192·27-s + 0.0195·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6019 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6019 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 + T \) |
| 463 | \( 1 + T \) |
| good | 2 | \( 1 + 2.12T + 2T^{2} \) |
| 3 | \( 1 + 0.167T + 3T^{2} \) |
| 5 | \( 1 + 1.92T + 5T^{2} \) |
| 7 | \( 1 - 0.0410T + 7T^{2} \) |
| 11 | \( 1 - 3.07T + 11T^{2} \) |
| 17 | \( 1 + 4.74T + 17T^{2} \) |
| 19 | \( 1 + 1.80T + 19T^{2} \) |
| 23 | \( 1 + 4.58T + 23T^{2} \) |
| 29 | \( 1 - 2.69T + 29T^{2} \) |
| 31 | \( 1 - 8.90T + 31T^{2} \) |
| 37 | \( 1 - 7.81T + 37T^{2} \) |
| 41 | \( 1 - 9.02T + 41T^{2} \) |
| 43 | \( 1 - 5.06T + 43T^{2} \) |
| 47 | \( 1 - 12.4T + 47T^{2} \) |
| 53 | \( 1 + 1.95T + 53T^{2} \) |
| 59 | \( 1 + 14.0T + 59T^{2} \) |
| 61 | \( 1 + 0.519T + 61T^{2} \) |
| 67 | \( 1 + 14.4T + 67T^{2} \) |
| 71 | \( 1 + 2.74T + 71T^{2} \) |
| 73 | \( 1 + 3.91T + 73T^{2} \) |
| 79 | \( 1 - 12.3T + 79T^{2} \) |
| 83 | \( 1 + 2.28T + 83T^{2} \) |
| 89 | \( 1 + 6.45T + 89T^{2} \) |
| 97 | \( 1 - 5.74T + 97T^{2} \) |
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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
−7.84742138744606065077465568453, −7.39877289925198770102948885970, −6.31199994524548568596099808599, −6.11654774801431226497860055255, −4.50210601204637262861753236490, −4.24001251288003409603691596836, −2.90433606046284658859208496982, −2.14479504089447348328035715652, −0.909686642338020304936742296298, 0,
0.909686642338020304936742296298, 2.14479504089447348328035715652, 2.90433606046284658859208496982, 4.24001251288003409603691596836, 4.50210601204637262861753236490, 6.11654774801431226497860055255, 6.31199994524548568596099808599, 7.39877289925198770102948885970, 7.84742138744606065077465568453
