L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s − 2·7-s + 8-s + 9-s + 10-s + 2·11-s − 12-s − 13-s − 2·14-s − 15-s + 16-s − 2·17-s + 18-s − 6·19-s + 20-s + 2·21-s + 2·22-s − 24-s + 25-s − 26-s − 27-s − 2·28-s + 6·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.603·11-s − 0.288·12-s − 0.277·13-s − 0.534·14-s − 0.258·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s − 1.37·19-s + 0.223·20-s + 0.436·21-s + 0.426·22-s − 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s − 0.377·28-s + 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 374790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 374790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.783660121\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.783660121\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−12.47595067908253, −12.20828858102488, −11.69543265833091, −11.10578518733998, −10.75046813876756, −10.37038679733805, −9.862637979725747, −9.338669715574160, −9.028616662176688, −8.295337255679982, −7.948476194084063, −7.074300523442910, −6.788282362144351, −6.439197131957513, −6.045855814933948, −5.551301353070115, −4.972697425717017, −4.493829002278172, −4.039065234855325, −3.572468627493694, −2.876540125110780, −2.260196021388133, −1.979373285244520, −0.9904485271940382, −0.5162863568791930,
0.5162863568791930, 0.9904485271940382, 1.979373285244520, 2.260196021388133, 2.876540125110780, 3.572468627493694, 4.039065234855325, 4.493829002278172, 4.972697425717017, 5.551301353070115, 6.045855814933948, 6.439197131957513, 6.788282362144351, 7.074300523442910, 7.948476194084063, 8.295337255679982, 9.028616662176688, 9.338669715574160, 9.862637979725747, 10.37038679733805, 10.75046813876756, 11.10578518733998, 11.69543265833091, 12.20828858102488, 12.47595067908253