| L(s) = 1 | + 2-s + 2·3-s − 2·4-s − 3·5-s + 2·6-s + 6·7-s − 3·8-s + 3·9-s − 3·10-s − 4·12-s + 8·13-s + 6·14-s − 6·15-s + 16-s + 17-s + 3·18-s + 5·19-s + 6·20-s + 12·21-s − 2·23-s − 6·24-s − 2·25-s + 8·26-s + 4·27-s − 12·28-s − 6·30-s − 31-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s − 4-s − 1.34·5-s + 0.816·6-s + 2.26·7-s − 1.06·8-s + 9-s − 0.948·10-s − 1.15·12-s + 2.21·13-s + 1.60·14-s − 1.54·15-s + 1/4·16-s + 0.242·17-s + 0.707·18-s + 1.14·19-s + 1.34·20-s + 2.61·21-s − 0.417·23-s − 1.22·24-s − 2/5·25-s + 1.56·26-s + 0.769·27-s − 2.26·28-s − 1.09·30-s − 0.179·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.901707972\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.901707972\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 3 T + 11 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 8 T + 37 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - T + 33 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 5 T + 33 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T + 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + T + 51 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 73 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 97 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - T + 93 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 17 T + 177 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 5 T + 63 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 9 T + 131 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - T + 33 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 9 T + 61 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 2 T + 142 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 10 T + 163 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 12 T + 157 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 10 T + 183 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - T - 17 T^{2} - p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.50398749116343667739074270250, −11.42839163980825173977288130985, −10.89581250907366373252466324033, −10.44187354639346064627124619476, −9.439131134488799967977287715585, −9.406146736517919592281054524453, −8.513070950631042094322441331278, −8.396210739694550870956355430360, −8.030904019089371728678372939174, −7.79526601890347810468062167552, −7.16710424225055155957688020296, −6.33312158733603880861521065816, −5.41772466495137110744896176397, −5.23332069893214574001633976046, −4.35999116674849621885614761016, −4.21280858197488992738118398970, −3.58177656039584076646750252854, −3.32008581174686335844406129445, −1.92517922199044961747531354354, −1.18710578751622857863542613928,
1.18710578751622857863542613928, 1.92517922199044961747531354354, 3.32008581174686335844406129445, 3.58177656039584076646750252854, 4.21280858197488992738118398970, 4.35999116674849621885614761016, 5.23332069893214574001633976046, 5.41772466495137110744896176397, 6.33312158733603880861521065816, 7.16710424225055155957688020296, 7.79526601890347810468062167552, 8.030904019089371728678372939174, 8.396210739694550870956355430360, 8.513070950631042094322441331278, 9.406146736517919592281054524453, 9.439131134488799967977287715585, 10.44187354639346064627124619476, 10.89581250907366373252466324033, 11.42839163980825173977288130985, 11.50398749116343667739074270250
