| L(s) = 1 | − 1.55·2-s − 1.14·3-s + 0.403·4-s + 5-s + 1.76·6-s + 3.74·7-s + 2.47·8-s − 1.69·9-s − 1.55·10-s + 1.45·11-s − 0.460·12-s − 6.71·13-s − 5.80·14-s − 1.14·15-s − 4.64·16-s + 2.63·18-s + 1.87·19-s + 0.403·20-s − 4.26·21-s − 2.26·22-s − 4.23·23-s − 2.82·24-s + 25-s + 10.4·26-s + 5.35·27-s + 1.51·28-s − 7.59·29-s + ⋯ |
| L(s) = 1 | − 1.09·2-s − 0.658·3-s + 0.201·4-s + 0.447·5-s + 0.721·6-s + 1.41·7-s + 0.874·8-s − 0.566·9-s − 0.490·10-s + 0.439·11-s − 0.132·12-s − 1.86·13-s − 1.55·14-s − 0.294·15-s − 1.16·16-s + 0.621·18-s + 0.430·19-s + 0.0902·20-s − 0.931·21-s − 0.482·22-s − 0.882·23-s − 0.576·24-s + 0.200·25-s + 2.04·26-s + 1.03·27-s + 0.285·28-s − 1.41·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 1.55T + 2T^{2} \) |
| 3 | \( 1 + 1.14T + 3T^{2} \) |
| 7 | \( 1 - 3.74T + 7T^{2} \) |
| 11 | \( 1 - 1.45T + 11T^{2} \) |
| 13 | \( 1 + 6.71T + 13T^{2} \) |
| 19 | \( 1 - 1.87T + 19T^{2} \) |
| 23 | \( 1 + 4.23T + 23T^{2} \) |
| 29 | \( 1 + 7.59T + 29T^{2} \) |
| 31 | \( 1 - 1.29T + 31T^{2} \) |
| 37 | \( 1 + 4.51T + 37T^{2} \) |
| 41 | \( 1 + 1.57T + 41T^{2} \) |
| 43 | \( 1 - 6.46T + 43T^{2} \) |
| 47 | \( 1 - 2.36T + 47T^{2} \) |
| 53 | \( 1 + 6.01T + 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 - 3.14T + 61T^{2} \) |
| 67 | \( 1 + 6.08T + 67T^{2} \) |
| 71 | \( 1 - 3.83T + 71T^{2} \) |
| 73 | \( 1 + 14.0T + 73T^{2} \) |
| 79 | \( 1 + 2.39T + 79T^{2} \) |
| 83 | \( 1 + 16.8T + 83T^{2} \) |
| 89 | \( 1 - 1.59T + 89T^{2} \) |
| 97 | \( 1 + 13.6T + 97T^{2} \) |
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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
−9.166383412323228217932176896281, −8.383275215070340390445362122853, −7.63477574469877146294484733000, −7.00311629928949843721376588815, −5.66622672079860542870841317219, −5.08493604792785141233318376687, −4.26150373106332913363930217921, −2.41771968299739893243985036794, −1.45623913074766883830594289825, 0,
1.45623913074766883830594289825, 2.41771968299739893243985036794, 4.26150373106332913363930217921, 5.08493604792785141233318376687, 5.66622672079860542870841317219, 7.00311629928949843721376588815, 7.63477574469877146294484733000, 8.383275215070340390445362122853, 9.166383412323228217932176896281
