| 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 + 6·13-s + 4·14-s + 2·15-s + 16-s − 2·17-s − 18-s − 4·19-s − 2·20-s + 4·21-s + 22-s − 4·23-s + 24-s − 25-s − 6·26-s − 27-s − 4·28-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 + 1.66·13-s + 1.06·14-s + 0.516·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.917·19-s − 0.447·20-s + 0.872·21-s + 0.213·22-s − 0.834·23-s + 0.204·24-s − 1/5·25-s − 1.17·26-s − 0.192·27-s − 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 37 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−14.03064791469779, −13.24467362222937, −13.17166975954890, −12.65522754133030, −12.00042124055811, −11.50497436532762, −11.20091775983383, −10.63039279749379, −10.11457154703051, −9.773585180178147, −9.055651231337427, −8.582503185125050, −8.185101051510283, −7.540036069324556, −6.978721090050212, −6.393309603805069, −6.174783019388790, −5.623233969449995, −4.720661808951502, −4.007966957556991, −3.549627468554636, −3.183690230946590, −2.147014246160107, −1.541536426217642, −0.4828173070382700, 0,
0.4828173070382700, 1.541536426217642, 2.147014246160107, 3.183690230946590, 3.549627468554636, 4.007966957556991, 4.720661808951502, 5.623233969449995, 6.174783019388790, 6.393309603805069, 6.978721090050212, 7.540036069324556, 8.185101051510283, 8.582503185125050, 9.055651231337427, 9.773585180178147, 10.11457154703051, 10.63039279749379, 11.20091775983383, 11.50497436532762, 12.00042124055811, 12.65522754133030, 13.17166975954890, 13.24467362222937, 14.03064791469779
