| L(s) = 1 | − 3-s − 4-s + 5-s − 3·7-s − 9-s + 3·11-s + 12-s − 6·13-s − 15-s + 16-s + 17-s + 19-s − 20-s + 3·21-s − 7·23-s − 2·25-s + 3·28-s − 29-s − 3·31-s − 3·33-s − 3·35-s + 36-s − 6·37-s + 6·39-s − 3·41-s − 3·44-s − 45-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1/2·4-s + 0.447·5-s − 1.13·7-s − 1/3·9-s + 0.904·11-s + 0.288·12-s − 1.66·13-s − 0.258·15-s + 1/4·16-s + 0.242·17-s + 0.229·19-s − 0.223·20-s + 0.654·21-s − 1.45·23-s − 2/5·25-s + 0.566·28-s − 0.185·29-s − 0.538·31-s − 0.522·33-s − 0.507·35-s + 1/6·36-s − 0.986·37-s + 0.960·39-s − 0.468·41-s − 0.452·44-s − 0.149·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26036 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26036 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + T^{2} \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 6 T + p T^{2} ) \) |
| 283 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 28 T + p T^{2} ) \) |
| good | 3 | $D_{4}$ | \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 3 T + 15 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 3 T + 13 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $D_{4}$ | \( 1 - T + 31 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 29 | $D_{4}$ | \( 1 + T + 43 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 3 T + 47 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 6 T + 46 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 3 T - 35 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $D_{4}$ | \( 1 - 14 T + 142 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 2 T + 16 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 5 T + 102 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $D_{4}$ | \( 1 + 9 T + 86 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + T + 76 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 9 T + 33 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 12 T + 196 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
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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
−15.7410676682, −15.1701100255, −14.6595496280, −14.2398937220, −13.8543306868, −13.3393962270, −12.8166700918, −12.2733350575, −11.9421148155, −11.6555712557, −10.7889481412, −10.1187808862, −9.92002293266, −9.42302756285, −8.99893984988, −8.27816487364, −7.54379444071, −6.98967004632, −6.43680919841, −5.70612911758, −5.47933913678, −4.54267716828, −3.84176119288, −3.05063345518, −1.96201705777, 0,
1.96201705777, 3.05063345518, 3.84176119288, 4.54267716828, 5.47933913678, 5.70612911758, 6.43680919841, 6.98967004632, 7.54379444071, 8.27816487364, 8.99893984988, 9.42302756285, 9.92002293266, 10.1187808862, 10.7889481412, 11.6555712557, 11.9421148155, 12.2733350575, 12.8166700918, 13.3393962270, 13.8543306868, 14.2398937220, 14.6595496280, 15.1701100255, 15.7410676682
