| L(s) = 1 | + 0.0114·2-s + 3-s − 1.99·4-s − 3.60·5-s + 0.0114·6-s + 7-s − 0.0458·8-s + 9-s − 0.0413·10-s − 6.22·11-s − 1.99·12-s − 0.693·13-s + 0.0114·14-s − 3.60·15-s + 3.99·16-s − 0.904·17-s + 0.0114·18-s + 6.54·19-s + 7.20·20-s + 21-s − 0.0713·22-s − 2.32·23-s − 0.0458·24-s + 7.98·25-s − 0.00794·26-s + 27-s − 1.99·28-s + ⋯ |
| L(s) = 1 | + 0.00810·2-s + 0.577·3-s − 0.999·4-s − 1.61·5-s + 0.00467·6-s + 0.377·7-s − 0.0162·8-s + 0.333·9-s − 0.0130·10-s − 1.87·11-s − 0.577·12-s − 0.192·13-s + 0.00306·14-s − 0.930·15-s + 0.999·16-s − 0.219·17-s + 0.00270·18-s + 1.50·19-s + 1.61·20-s + 0.218·21-s − 0.0152·22-s − 0.484·23-s − 0.00935·24-s + 1.59·25-s − 0.00155·26-s + 0.192·27-s − 0.377·28-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 383 | \( 1 + T \) |
| good | 2 | \( 1 - 0.0114T + 2T^{2} \) |
| 5 | \( 1 + 3.60T + 5T^{2} \) |
| 11 | \( 1 + 6.22T + 11T^{2} \) |
| 13 | \( 1 + 0.693T + 13T^{2} \) |
| 17 | \( 1 + 0.904T + 17T^{2} \) |
| 19 | \( 1 - 6.54T + 19T^{2} \) |
| 23 | \( 1 + 2.32T + 23T^{2} \) |
| 29 | \( 1 - 5.92T + 29T^{2} \) |
| 31 | \( 1 + 7.75T + 31T^{2} \) |
| 37 | \( 1 - 5.43T + 37T^{2} \) |
| 41 | \( 1 - 11.0T + 41T^{2} \) |
| 43 | \( 1 - 8.88T + 43T^{2} \) |
| 47 | \( 1 + 8.78T + 47T^{2} \) |
| 53 | \( 1 + 4.91T + 53T^{2} \) |
| 59 | \( 1 - 1.38T + 59T^{2} \) |
| 61 | \( 1 + 3.79T + 61T^{2} \) |
| 67 | \( 1 - 4.62T + 67T^{2} \) |
| 71 | \( 1 + 4.03T + 71T^{2} \) |
| 73 | \( 1 - 6.24T + 73T^{2} \) |
| 79 | \( 1 + 11.6T + 79T^{2} \) |
| 83 | \( 1 - 3.52T + 83T^{2} \) |
| 89 | \( 1 - 14.3T + 89T^{2} \) |
| 97 | \( 1 - 7.25T + 97T^{2} \) |
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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
−7.68455568599886015503000078080, −7.35106678316192796428116011725, −5.93316398545539305480037962437, −5.07260728472175996405654315832, −4.65438244126688659269870551559, −3.91088152997222511648276989028, −3.21412758912702367173529882910, −2.50989191974302433365921594043, −0.958781772417229425392744427056, 0,
0.958781772417229425392744427056, 2.50989191974302433365921594043, 3.21412758912702367173529882910, 3.91088152997222511648276989028, 4.65438244126688659269870551559, 5.07260728472175996405654315832, 5.93316398545539305480037962437, 7.35106678316192796428116011725, 7.68455568599886015503000078080
