L(s) = 1 | + 2-s − 3-s + 4-s + 2·5-s − 6-s + 4·7-s + 8-s + 9-s + 2·10-s + 11-s − 12-s − 4·13-s + 4·14-s − 2·15-s + 16-s + 18-s − 8·19-s + 2·20-s − 4·21-s + 22-s − 24-s − 25-s − 4·26-s − 27-s + 4·28-s − 2·30-s − 10·31-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s + 1.51·7-s + 0.353·8-s + 1/3·9-s + 0.632·10-s + 0.301·11-s − 0.288·12-s − 1.10·13-s + 1.06·14-s − 0.516·15-s + 1/4·16-s + 0.235·18-s − 1.83·19-s + 0.447·20-s − 0.872·21-s + 0.213·22-s − 0.204·24-s − 1/5·25-s − 0.784·26-s − 0.192·27-s + 0.755·28-s − 0.365·30-s − 1.79·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19074 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19074 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−15.92435305180826, −15.25953559903750, −14.73540044173146, −14.38441916270977, −13.97610673142036, −13.23128001090247, −12.55619582781955, −12.36539461376133, −11.53720959110764, −11.07587029269359, −10.61734158458755, −10.08410885720174, −9.274363714260776, −8.724533450183503, −7.901144872405312, −7.353957884656157, −6.724898861698675, −6.002285438101875, −5.520110347795975, −4.923388887671205, −4.442883185632045, −3.784805151837642, −2.562103014599905, −1.913484352984504, −1.523914228386670, 0,
1.523914228386670, 1.913484352984504, 2.562103014599905, 3.784805151837642, 4.442883185632045, 4.923388887671205, 5.520110347795975, 6.002285438101875, 6.724898861698675, 7.353957884656157, 7.901144872405312, 8.724533450183503, 9.274363714260776, 10.08410885720174, 10.61734158458755, 11.07587029269359, 11.53720959110764, 12.36539461376133, 12.55619582781955, 13.23128001090247, 13.97610673142036, 14.38441916270977, 14.73540044173146, 15.25953559903750, 15.92435305180826