L(s) = 1 | + 2·2-s + 2·4-s + 7-s − 4·13-s + 2·14-s − 4·16-s − 17-s + 4·23-s − 8·26-s + 2·28-s − 2·31-s − 8·32-s − 2·34-s − 6·37-s − 2·41-s − 3·43-s + 8·46-s + 7·47-s + 49-s − 8·52-s + 12·53-s + 5·59-s + 12·61-s − 4·62-s − 8·64-s − 5·67-s − 2·68-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 4-s + 0.377·7-s − 1.10·13-s + 0.534·14-s − 16-s − 0.242·17-s + 0.834·23-s − 1.56·26-s + 0.377·28-s − 0.359·31-s − 1.41·32-s − 0.342·34-s − 0.986·37-s − 0.312·41-s − 0.457·43-s + 1.17·46-s + 1.02·47-s + 1/7·49-s − 1.10·52-s + 1.64·53-s + 0.650·59-s + 1.53·61-s − 0.508·62-s − 64-s − 0.610·67-s − 0.242·68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 190575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - p T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.33909019534064, −12.91003351850597, −12.48162144841160, −12.02531269695741, −11.68577853451431, −11.16626563300056, −10.70499571228186, −10.09565219810505, −9.663932500030065, −8.942131310867916, −8.694788422835109, −8.017126419644571, −7.340705251074261, −6.830327533773991, −6.746979785887504, −5.716385372489230, −5.430019616656857, −5.083819252397066, −4.452370593276558, −4.047218302237927, −3.478673186300093, −2.838382393266623, −2.363369330093071, −1.834223938247874, −0.8695746383248621, 0,
0.8695746383248621, 1.834223938247874, 2.363369330093071, 2.838382393266623, 3.478673186300093, 4.047218302237927, 4.452370593276558, 5.083819252397066, 5.430019616656857, 5.716385372489230, 6.746979785887504, 6.830327533773991, 7.340705251074261, 8.017126419644571, 8.694788422835109, 8.942131310867916, 9.663932500030065, 10.09565219810505, 10.70499571228186, 11.16626563300056, 11.68577853451431, 12.02531269695741, 12.48162144841160, 12.91003351850597, 13.33909019534064