| L(s) = 1 | − 2·2-s + 4-s − 4·7-s − 4·11-s + 2·13-s + 8·14-s + 16-s − 4·17-s + 4·19-s + 8·22-s − 4·26-s − 4·28-s + 12·31-s + 2·32-s + 8·34-s − 8·38-s + 12·41-s + 8·43-s − 4·44-s − 4·47-s + 6·49-s + 2·52-s − 12·53-s − 12·59-s − 16·61-s − 24·62-s − 11·64-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1/2·4-s − 1.51·7-s − 1.20·11-s + 0.554·13-s + 2.13·14-s + 1/4·16-s − 0.970·17-s + 0.917·19-s + 1.70·22-s − 0.784·26-s − 0.755·28-s + 2.15·31-s + 0.353·32-s + 1.37·34-s − 1.29·38-s + 1.87·41-s + 1.21·43-s − 0.603·44-s − 0.583·47-s + 6/7·49-s + 0.277·52-s − 1.64·53-s − 1.56·59-s − 2.04·61-s − 3.04·62-s − 1.37·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8555625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8555625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | $C_4$ | \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 24 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 40 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 12 T + 80 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 52 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 90 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 12 T + 70 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 12 T + 136 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 4 T + 48 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 12 T + 194 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 166 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−8.507851675871047777054359210651, −8.275731186737373112759147165152, −7.889392780197851197501579569338, −7.71732947983134690111653698065, −7.12380995133073876687621977894, −6.74903189495447675729284921469, −6.32498762067638238539298046728, −6.00436687902283535343009165114, −5.77191896227545300030080996378, −5.10917826261536785364567787512, −4.54358292044570348135193158744, −4.33972661804074621763948430631, −3.69942047285066322199294326367, −2.99453290549644634905242411587, −2.76739397170308285048990857327, −2.60389878120004847109658390383, −1.44217323418309889110440269678, −1.03949577710909889928289313244, 0, 0,
1.03949577710909889928289313244, 1.44217323418309889110440269678, 2.60389878120004847109658390383, 2.76739397170308285048990857327, 2.99453290549644634905242411587, 3.69942047285066322199294326367, 4.33972661804074621763948430631, 4.54358292044570348135193158744, 5.10917826261536785364567787512, 5.77191896227545300030080996378, 6.00436687902283535343009165114, 6.32498762067638238539298046728, 6.74903189495447675729284921469, 7.12380995133073876687621977894, 7.71732947983134690111653698065, 7.889392780197851197501579569338, 8.275731186737373112759147165152, 8.507851675871047777054359210651
