| L(s) = 1 | + 2.61·2-s + 0.929·3-s + 4.85·4-s − 5-s + 2.43·6-s − 4.67·7-s + 7.46·8-s − 2.13·9-s − 2.61·10-s + 11-s + 4.50·12-s − 6.33·13-s − 12.2·14-s − 0.929·15-s + 9.83·16-s − 5.62·17-s − 5.59·18-s − 3.70·19-s − 4.85·20-s − 4.34·21-s + 2.61·22-s + 7.36·23-s + 6.93·24-s + 25-s − 16.5·26-s − 4.77·27-s − 22.6·28-s + ⋯ |
| L(s) = 1 | + 1.85·2-s + 0.536·3-s + 2.42·4-s − 0.447·5-s + 0.993·6-s − 1.76·7-s + 2.63·8-s − 0.711·9-s − 0.827·10-s + 0.301·11-s + 1.30·12-s − 1.75·13-s − 3.26·14-s − 0.240·15-s + 2.45·16-s − 1.36·17-s − 1.31·18-s − 0.849·19-s − 1.08·20-s − 0.947·21-s + 0.558·22-s + 1.53·23-s + 1.41·24-s + 0.200·25-s − 3.25·26-s − 0.918·27-s − 4.28·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| good | 2 | \( 1 - 2.61T + 2T^{2} \) |
| 3 | \( 1 - 0.929T + 3T^{2} \) |
| 7 | \( 1 + 4.67T + 7T^{2} \) |
| 13 | \( 1 + 6.33T + 13T^{2} \) |
| 17 | \( 1 + 5.62T + 17T^{2} \) |
| 19 | \( 1 + 3.70T + 19T^{2} \) |
| 23 | \( 1 - 7.36T + 23T^{2} \) |
| 29 | \( 1 - 0.441T + 29T^{2} \) |
| 31 | \( 1 - 4.72T + 31T^{2} \) |
| 37 | \( 1 + 2.76T + 37T^{2} \) |
| 41 | \( 1 - 4.91T + 41T^{2} \) |
| 43 | \( 1 - 4.27T + 43T^{2} \) |
| 47 | \( 1 + 9.12T + 47T^{2} \) |
| 53 | \( 1 - 6.25T + 53T^{2} \) |
| 59 | \( 1 + 13.2T + 59T^{2} \) |
| 61 | \( 1 + 15.2T + 61T^{2} \) |
| 67 | \( 1 + 14.3T + 67T^{2} \) |
| 71 | \( 1 - 4.16T + 71T^{2} \) |
| 79 | \( 1 - 9.32T + 79T^{2} \) |
| 83 | \( 1 - 0.533T + 83T^{2} \) |
| 89 | \( 1 + 8.81T + 89T^{2} \) |
| 97 | \( 1 - 4.67T + 97T^{2} \) |
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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
−7.70633795518469833295702932629, −6.98756820300212842955464563924, −6.51521437722386139490585651580, −5.87251002984046721448497069072, −4.79395992958687955296582235841, −4.33438644540791217760667790872, −3.28270629548680446078458626445, −2.88845396965530018628191379263, −2.25283667720101175597292524254, 0,
2.25283667720101175597292524254, 2.88845396965530018628191379263, 3.28270629548680446078458626445, 4.33438644540791217760667790872, 4.79395992958687955296582235841, 5.87251002984046721448497069072, 6.51521437722386139490585651580, 6.98756820300212842955464563924, 7.70633795518469833295702932629
