| L(s) = 1 | + 0.0672·2-s − 1.99·4-s − 5-s + 2.46·7-s − 0.268·8-s − 0.0672·10-s + 3.21·11-s − 13-s + 0.165·14-s + 3.97·16-s − 4.77·17-s − 3.94·19-s + 1.99·20-s + 0.215·22-s + 4.26·23-s + 25-s − 0.0672·26-s − 4.91·28-s − 2.30·29-s − 7.63·31-s + 0.803·32-s − 0.321·34-s − 2.46·35-s + 4.87·37-s − 0.265·38-s + 0.268·40-s − 2.53·41-s + ⋯ |
| L(s) = 1 | + 0.0475·2-s − 0.997·4-s − 0.447·5-s + 0.931·7-s − 0.0949·8-s − 0.0212·10-s + 0.968·11-s − 0.277·13-s + 0.0442·14-s + 0.993·16-s − 1.15·17-s − 0.905·19-s + 0.446·20-s + 0.0460·22-s + 0.890·23-s + 0.200·25-s − 0.0131·26-s − 0.929·28-s − 0.428·29-s − 1.37·31-s + 0.142·32-s − 0.0550·34-s − 0.416·35-s + 0.801·37-s − 0.0430·38-s + 0.0424·40-s − 0.395·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5265 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - 0.0672T + 2T^{2} \) |
| 7 | \( 1 - 2.46T + 7T^{2} \) |
| 11 | \( 1 - 3.21T + 11T^{2} \) |
| 17 | \( 1 + 4.77T + 17T^{2} \) |
| 19 | \( 1 + 3.94T + 19T^{2} \) |
| 23 | \( 1 - 4.26T + 23T^{2} \) |
| 29 | \( 1 + 2.30T + 29T^{2} \) |
| 31 | \( 1 + 7.63T + 31T^{2} \) |
| 37 | \( 1 - 4.87T + 37T^{2} \) |
| 41 | \( 1 + 2.53T + 41T^{2} \) |
| 43 | \( 1 + 4.56T + 43T^{2} \) |
| 47 | \( 1 + 6.26T + 47T^{2} \) |
| 53 | \( 1 - 12.4T + 53T^{2} \) |
| 59 | \( 1 - 2.85T + 59T^{2} \) |
| 61 | \( 1 - 12.7T + 61T^{2} \) |
| 67 | \( 1 + 13.5T + 67T^{2} \) |
| 71 | \( 1 - 7.79T + 71T^{2} \) |
| 73 | \( 1 - 2.26T + 73T^{2} \) |
| 79 | \( 1 - 9.13T + 79T^{2} \) |
| 83 | \( 1 - 10.5T + 83T^{2} \) |
| 89 | \( 1 - 0.966T + 89T^{2} \) |
| 97 | \( 1 + 18.7T + 97T^{2} \) |
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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
−8.040885143661924325542276218046, −7.12637193910023367190129086257, −6.50548690009790743156730762448, −5.41795088258643504999776876059, −4.83214969475191608300341695843, −4.14182640050003278380265954015, −3.62064461945924408351489069216, −2.28925223267146091664076068331, −1.25933192489724060723888331640, 0,
1.25933192489724060723888331640, 2.28925223267146091664076068331, 3.62064461945924408351489069216, 4.14182640050003278380265954015, 4.83214969475191608300341695843, 5.41795088258643504999776876059, 6.50548690009790743156730762448, 7.12637193910023367190129086257, 8.040885143661924325542276218046
