| L(s) = 1 | + 2·2-s + 2·4-s + 7-s + 3·13-s + 2·14-s − 4·16-s − 3·17-s + 7·19-s − 4·23-s + 6·26-s + 2·28-s − 4·29-s + 2·31-s − 8·32-s − 6·34-s − 3·37-s + 14·38-s + 5·41-s − 4·43-s − 8·46-s − 8·47-s + 49-s + 6·52-s + 9·53-s − 8·58-s + 12·59-s + 9·61-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 0.377·7-s + 0.832·13-s + 0.534·14-s − 16-s − 0.727·17-s + 1.60·19-s − 0.834·23-s + 1.17·26-s + 0.377·28-s − 0.742·29-s + 0.359·31-s − 1.41·32-s − 1.02·34-s − 0.493·37-s + 2.27·38-s + 0.780·41-s − 0.609·43-s − 1.17·46-s − 1.16·47-s + 1/7·49-s + 0.832·52-s + 1.23·53-s − 1.05·58-s + 1.56·59-s + 1.15·61-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 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 9 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−13.27469430621343, −13.08189770689914, −12.54581517815052, −11.85195939486564, −11.56959208692290, −11.34710145474049, −10.73885092900688, −10.03895434620264, −9.653641002939258, −9.070723527963886, −8.428340288064048, −8.202265901623093, −7.358257588462577, −6.937575597319610, −6.505624210250272, −5.815422665669551, −5.424860939392872, −5.162723029623680, −4.339486344482448, −3.989275983858094, −3.555454170726025, −2.907370031152826, −2.363816745116450, −1.693455791991594, −0.9845133801070364, 0,
0.9845133801070364, 1.693455791991594, 2.363816745116450, 2.907370031152826, 3.555454170726025, 3.989275983858094, 4.339486344482448, 5.162723029623680, 5.424860939392872, 5.815422665669551, 6.505624210250272, 6.937575597319610, 7.358257588462577, 8.202265901623093, 8.428340288064048, 9.070723527963886, 9.653641002939258, 10.03895434620264, 10.73885092900688, 11.34710145474049, 11.56959208692290, 11.85195939486564, 12.54581517815052, 13.08189770689914, 13.27469430621343
