| L(s) = 1 | − 0.589·2-s − 1.65·4-s − 1.68·5-s + 4.90·7-s + 2.15·8-s + 0.995·10-s − 11-s − 1.98·13-s − 2.89·14-s + 2.03·16-s − 3.06·17-s + 2.30·19-s + 2.79·20-s + 0.589·22-s − 1.69·23-s − 2.14·25-s + 1.17·26-s − 8.10·28-s + 2.82·29-s + 9.71·31-s − 5.50·32-s + 1.80·34-s − 8.28·35-s − 5.86·37-s − 1.36·38-s − 3.63·40-s − 6.61·41-s + ⋯ |
| L(s) = 1 | − 0.416·2-s − 0.826·4-s − 0.755·5-s + 1.85·7-s + 0.761·8-s + 0.314·10-s − 0.301·11-s − 0.550·13-s − 0.772·14-s + 0.508·16-s − 0.742·17-s + 0.529·19-s + 0.623·20-s + 0.125·22-s − 0.352·23-s − 0.429·25-s + 0.229·26-s − 1.53·28-s + 0.525·29-s + 1.74·31-s − 0.973·32-s + 0.309·34-s − 1.39·35-s − 0.964·37-s − 0.220·38-s − 0.575·40-s − 1.03·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8019 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8019 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + 0.589T + 2T^{2} \) |
| 5 | \( 1 + 1.68T + 5T^{2} \) |
| 7 | \( 1 - 4.90T + 7T^{2} \) |
| 13 | \( 1 + 1.98T + 13T^{2} \) |
| 17 | \( 1 + 3.06T + 17T^{2} \) |
| 19 | \( 1 - 2.30T + 19T^{2} \) |
| 23 | \( 1 + 1.69T + 23T^{2} \) |
| 29 | \( 1 - 2.82T + 29T^{2} \) |
| 31 | \( 1 - 9.71T + 31T^{2} \) |
| 37 | \( 1 + 5.86T + 37T^{2} \) |
| 41 | \( 1 + 6.61T + 41T^{2} \) |
| 43 | \( 1 - 3.46T + 43T^{2} \) |
| 47 | \( 1 - 11.8T + 47T^{2} \) |
| 53 | \( 1 + 12.0T + 53T^{2} \) |
| 59 | \( 1 + 7.78T + 59T^{2} \) |
| 61 | \( 1 + 6.93T + 61T^{2} \) |
| 67 | \( 1 + 15.3T + 67T^{2} \) |
| 71 | \( 1 + 0.419T + 71T^{2} \) |
| 73 | \( 1 + 2.45T + 73T^{2} \) |
| 79 | \( 1 + 5.68T + 79T^{2} \) |
| 83 | \( 1 + 7.24T + 83T^{2} \) |
| 89 | \( 1 - 15.6T + 89T^{2} \) |
| 97 | \( 1 - 11.7T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.71505391289041709161705436835, −7.20546435508330871620918867559, −6.00096964261398380853968299741, −5.12509100572265726737183017376, −4.54307441474468186545745867525, −4.30103887170276752308802377310, −3.12681547819548995520690919947, −2.00256786149898099942161316087, −1.14653184036494723530230915009, 0,
1.14653184036494723530230915009, 2.00256786149898099942161316087, 3.12681547819548995520690919947, 4.30103887170276752308802377310, 4.54307441474468186545745867525, 5.12509100572265726737183017376, 6.00096964261398380853968299741, 7.20546435508330871620918867559, 7.71505391289041709161705436835
