| L(s) = 1 | + 2-s − 3-s + 4-s + 2·5-s − 6-s + 8-s + 9-s + 2·10-s − 12-s − 2·15-s + 16-s + 17-s + 18-s + 2·20-s − 24-s − 25-s − 27-s − 2·29-s − 2·30-s − 8·31-s + 32-s + 34-s + 36-s − 10·37-s + 2·40-s + 6·41-s − 4·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.632·10-s − 0.288·12-s − 0.516·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s + 0.447·20-s − 0.204·24-s − 1/5·25-s − 0.192·27-s − 0.371·29-s − 0.365·30-s − 1.43·31-s + 0.176·32-s + 0.171·34-s + 1/6·36-s − 1.64·37-s + 0.316·40-s + 0.937·41-s − 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17238 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17238 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−16.03099500350302, −15.68638674549437, −14.90413839543693, −14.44985103528364, −13.86223191088424, −13.40671968672524, −12.81968167327974, −12.35469515791764, −11.79395317473320, −11.10923901542086, −10.64861805006636, −10.09312602096399, −9.439338601022648, −8.922059118657555, −7.997017655677878, −7.318075434781398, −6.775995045086682, −6.081408272752196, −5.554776587883092, −5.197205320681253, −4.326473710352203, −3.667378205823021, −2.855461538480467, −1.942947901416895, −1.401560756832267, 0,
1.401560756832267, 1.942947901416895, 2.855461538480467, 3.667378205823021, 4.326473710352203, 5.197205320681253, 5.554776587883092, 6.081408272752196, 6.775995045086682, 7.318075434781398, 7.997017655677878, 8.922059118657555, 9.439338601022648, 10.09312602096399, 10.64861805006636, 11.10923901542086, 11.79395317473320, 12.35469515791764, 12.81968167327974, 13.40671968672524, 13.86223191088424, 14.44985103528364, 14.90413839543693, 15.68638674549437, 16.03099500350302
