| L(s) = 1 | − 8·2-s + 14·3-s + 48·4-s + 42·5-s − 112·6-s − 256·8-s + 181·9-s − 336·10-s − 294·11-s + 672·12-s + 140·13-s + 588·15-s + 1.28e3·16-s − 1.30e3·17-s − 1.44e3·18-s + 1.44e3·19-s + 2.01e3·20-s + 2.35e3·22-s + 2.64e3·23-s − 3.58e3·24-s − 4.92e3·25-s − 1.12e3·26-s + 5.72e3·27-s + 1.66e3·29-s − 4.70e3·30-s + 1.47e4·31-s − 6.14e3·32-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.898·3-s + 3/2·4-s + 0.751·5-s − 1.27·6-s − 1.41·8-s + 0.744·9-s − 1.06·10-s − 0.732·11-s + 1.34·12-s + 0.229·13-s + 0.674·15-s + 5/4·16-s − 1.09·17-s − 1.05·18-s + 0.916·19-s + 1.12·20-s + 1.03·22-s + 1.04·23-s − 1.27·24-s − 1.57·25-s − 0.324·26-s + 1.51·27-s + 0.368·29-s − 0.954·30-s + 2.76·31-s − 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.127377765\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.127377765\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 - 14 T + 5 p T^{2} - 14 p^{5} T^{3} + p^{10} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 21 T + p^{5} T^{2} )^{2} \) |
| 11 | $D_{4}$ | \( 1 + 294 T + 114391 T^{2} + 294 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 140 T + 728766 T^{2} - 140 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 1302 T + 3095035 T^{2} + 1302 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 1442 T + 4119519 T^{2} - 1442 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2646 T + 8890015 T^{2} - 2646 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 1668 T + 40800574 T^{2} - 1668 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 14798 T + 109078503 T^{2} - 14798 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 5182 T + 71101515 T^{2} - 5182 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 5124 T + 23950966 T^{2} - 5124 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4520 T + 240418566 T^{2} + 4520 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 14994 T + 427056103 T^{2} - 14994 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 24006 T + 935516275 T^{2} - 24006 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 38850 T + 1324947343 T^{2} + 38850 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 23618 T + 1734277563 T^{2} - 23618 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 32002 T + 1139838495 T^{2} - 32002 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 89376 T + 5425689166 T^{2} + 89376 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 47138 T + 4432072947 T^{2} - 47138 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 40970 T + 6023150703 T^{2} + 40970 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 68376 T + 8908447510 T^{2} + 68376 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 123102 T + 13551221779 T^{2} + 123102 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 43652 T + 8867162790 T^{2} - 43652 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−13.28273545224890244975325750588, −12.81709534689572315232736598469, −11.95950668698245076315898920684, −11.56098116642759131289371596203, −10.83913495993302690137834504632, −10.24786076388569582905627734501, −9.898934573701429047103416194585, −9.486434220274517153296047473151, −8.637272735487623794522720967327, −8.589232309011873197609852457946, −7.74720702211958918746642925833, −7.31375942334279394980130482519, −6.53791483958374596983253057837, −6.02481892948640697300908545664, −5.04945301378655417418616814790, −4.12623272533962640301630900794, −2.71593181875707664145104247036, −2.67208281501642659223431370480, −1.52930606058188947772642361523, −0.71401889630299768595850059387,
0.71401889630299768595850059387, 1.52930606058188947772642361523, 2.67208281501642659223431370480, 2.71593181875707664145104247036, 4.12623272533962640301630900794, 5.04945301378655417418616814790, 6.02481892948640697300908545664, 6.53791483958374596983253057837, 7.31375942334279394980130482519, 7.74720702211958918746642925833, 8.589232309011873197609852457946, 8.637272735487623794522720967327, 9.486434220274517153296047473151, 9.898934573701429047103416194585, 10.24786076388569582905627734501, 10.83913495993302690137834504632, 11.56098116642759131289371596203, 11.95950668698245076315898920684, 12.81709534689572315232736598469, 13.28273545224890244975325750588
