| L(s) = 1 | + 2·2-s − 2·3-s + 2·4-s − 4·5-s − 4·6-s − 2·7-s + 4·8-s + 3·9-s − 8·10-s − 4·12-s + 2·13-s − 4·14-s + 8·15-s + 8·16-s − 4·17-s + 6·18-s + 6·19-s − 8·20-s + 4·21-s + 10·23-s − 8·24-s + 5·25-s + 4·26-s − 4·27-s − 4·28-s + 6·29-s + 16·30-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 4-s − 1.78·5-s − 1.63·6-s − 0.755·7-s + 1.41·8-s + 9-s − 2.52·10-s − 1.15·12-s + 0.554·13-s − 1.06·14-s + 2.06·15-s + 2·16-s − 0.970·17-s + 1.41·18-s + 1.37·19-s − 1.78·20-s + 0.872·21-s + 2.08·23-s − 1.63·24-s + 25-s + 0.784·26-s − 0.769·27-s − 0.755·28-s + 1.11·29-s + 2.92·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6456681 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6456681 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.650858668\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.650858668\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 24 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 35 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 6 T + 20 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 10 T + 68 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 6 T + 64 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 12 T + 98 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 + 6 T + 83 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 8 T + 83 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 12 T + 130 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 4 T - 25 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 6 T + 104 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 6 T + 4 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 10 T + 168 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 162 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 12 T + 199 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 12 T + 139 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 10 T + 216 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
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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
−8.806192853137549481826411549447, −8.676281707668627280735190554943, −8.381464303091072692591087458863, −7.47460746358181216177985236969, −7.41970180623759264440963663530, −7.27285948416964649393150172779, −6.59228837096630877548442198614, −6.50983749682833170224018873803, −5.88442055714005914450239116438, −5.47305015773542285744535752276, −4.91083117771367685384378725219, −4.86950747774797776140969758461, −4.24894808291035543973418875132, −4.17499346732336562873704834000, −3.39604296156803352527871372482, −3.30278121518021525712174392459, −2.81954108000418080807248526963, −1.80774748779950713604646221011, −1.04011128561946551079017709498, −0.57191706615010549654599890253,
0.57191706615010549654599890253, 1.04011128561946551079017709498, 1.80774748779950713604646221011, 2.81954108000418080807248526963, 3.30278121518021525712174392459, 3.39604296156803352527871372482, 4.17499346732336562873704834000, 4.24894808291035543973418875132, 4.86950747774797776140969758461, 4.91083117771367685384378725219, 5.47305015773542285744535752276, 5.88442055714005914450239116438, 6.50983749682833170224018873803, 6.59228837096630877548442198614, 7.27285948416964649393150172779, 7.41970180623759264440963663530, 7.47460746358181216177985236969, 8.381464303091072692591087458863, 8.676281707668627280735190554943, 8.806192853137549481826411549447
