| L(s) = 1 | − 2-s + 4-s − 2·7-s − 8-s − 3·11-s − 2·13-s + 2·14-s + 16-s + 3·17-s + 7·19-s + 3·22-s + 4·23-s + 2·26-s − 2·28-s + 6·29-s + 4·31-s − 32-s − 3·34-s − 4·37-s − 7·38-s − 7·41-s − 4·43-s − 3·44-s − 4·46-s + 10·47-s − 3·49-s − 2·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.755·7-s − 0.353·8-s − 0.904·11-s − 0.554·13-s + 0.534·14-s + 1/4·16-s + 0.727·17-s + 1.60·19-s + 0.639·22-s + 0.834·23-s + 0.392·26-s − 0.377·28-s + 1.11·29-s + 0.718·31-s − 0.176·32-s − 0.514·34-s − 0.657·37-s − 1.13·38-s − 1.09·41-s − 0.609·43-s − 0.452·44-s − 0.589·46-s + 1.45·47-s − 3/7·49-s − 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.513524501\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.513524501\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 167 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 15 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.91525119645892, −13.54866768291898, −13.18038975329739, −12.25624117745770, −12.11132340116979, −11.73002166442120, −10.80058635024664, −10.45387827170772, −10.02366491527374, −9.577825960842205, −9.072686046816612, −8.503476115711222, −7.794224443302800, −7.602152528036515, −6.761658231812063, −6.635381011575865, −5.614084064839775, −5.296479425599257, −4.753186109083122, −3.754834647951125, −3.073829398192578, −2.844499919129130, −2.021469180571614, −1.054334296229125, −0.5318563058479643,
0.5318563058479643, 1.054334296229125, 2.021469180571614, 2.844499919129130, 3.073829398192578, 3.754834647951125, 4.753186109083122, 5.296479425599257, 5.614084064839775, 6.635381011575865, 6.761658231812063, 7.602152528036515, 7.794224443302800, 8.503476115711222, 9.072686046816612, 9.577825960842205, 10.02366491527374, 10.45387827170772, 10.80058635024664, 11.73002166442120, 12.11132340116979, 12.25624117745770, 13.18038975329739, 13.54866768291898, 13.91525119645892
