| L(s) = 1 | − 2·2-s − 2·3-s + 4-s − 5·5-s + 4·6-s − 4·7-s + 9-s + 10·10-s − 6·11-s − 2·12-s + 8·14-s + 10·15-s + 16-s + 3·17-s − 2·18-s − 2·19-s − 5·20-s + 8·21-s + 12·22-s + 2·23-s + 16·25-s + 4·27-s − 4·28-s − 10·29-s − 20·30-s − 9·31-s + 2·32-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 1/2·4-s − 2.23·5-s + 1.63·6-s − 1.51·7-s + 1/3·9-s + 3.16·10-s − 1.80·11-s − 0.577·12-s + 2.13·14-s + 2.58·15-s + 1/4·16-s + 0.727·17-s − 0.471·18-s − 0.458·19-s − 1.11·20-s + 1.74·21-s + 2.55·22-s + 0.417·23-s + 16/5·25-s + 0.769·27-s − 0.755·28-s − 1.85·29-s − 3.65·30-s − 1.61·31-s + 0.353·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9585 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9585 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 12 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $D_{4}$ | \( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $D_{4}$ | \( 1 - 3 T - 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T - 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T - 16 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 10 T + 70 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 37 | $D_{4}$ | \( 1 + 9 T + 80 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 8 T + 62 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $D_{4}$ | \( 1 + 6 T + 64 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $D_{4}$ | \( 1 - T + 84 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 11 T + 116 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 128 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 89 | $C_4$ | \( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 2 T + 54 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−16.8822299219, −16.6555242833, −16.4161559030, −15.7752473258, −15.5069003206, −15.1083398821, −14.3875398548, −13.3081995632, −12.8338475570, −12.4421944097, −12.1226113296, −11.2519273183, −10.9786280684, −10.5660224349, −9.85747449866, −9.36393698070, −8.60912690418, −8.16676495295, −7.57021223666, −7.14177769827, −6.37823034201, −5.44948138270, −4.88110684836, −3.63455148318, −3.16656671454, 0, 0,
3.16656671454, 3.63455148318, 4.88110684836, 5.44948138270, 6.37823034201, 7.14177769827, 7.57021223666, 8.16676495295, 8.60912690418, 9.36393698070, 9.85747449866, 10.5660224349, 10.9786280684, 11.2519273183, 12.1226113296, 12.4421944097, 12.8338475570, 13.3081995632, 14.3875398548, 15.1083398821, 15.5069003206, 15.7752473258, 16.4161559030, 16.6555242833, 16.8822299219
