| L(s) = 1 | + 52·9-s + 52·11-s + 296·23-s + 242·25-s − 472·29-s + 508·37-s + 488·43-s + 340·53-s − 840·67-s + 1.68e3·71-s − 2.10e3·79-s + 729·81-s + 2.70e3·99-s − 1.76e3·107-s − 1.19e3·109-s − 6.19e3·113-s + 3.33e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 6.48e3·169-s + ⋯ |
| L(s) = 1 | + 1.92·9-s + 1.42·11-s + 2.68·23-s + 1.93·25-s − 3.02·29-s + 2.25·37-s + 1.73·43-s + 0.881·53-s − 1.53·67-s + 2.80·71-s − 2.99·79-s + 81-s + 2.74·99-s − 1.59·107-s − 1.05·109-s − 5.15·113-s + 2.50·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s − 2.95·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(7.435937223\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.435937223\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^3$ | \( 1 - 52 T^{2} + 1975 T^{4} - 52 p^{6} T^{6} + p^{12} T^{8} \) |
| 5 | $C_2^3$ | \( 1 - 242 T^{2} + 42939 T^{4} - 242 p^{6} T^{6} + p^{12} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - 26 T - 655 T^{2} - 26 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 3242 T^{2} + p^{6} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 + 832 T^{2} - 23445345 T^{4} + 832 p^{6} T^{6} + p^{12} T^{8} \) |
| 19 | $C_2^3$ | \( 1 - 4740 T^{2} - 24578281 T^{4} - 4740 p^{6} T^{6} + p^{12} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 - 148 T + 9737 T^{2} - 148 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 118 T + p^{3} T^{2} )^{4} \) |
| 31 | $C_2^3$ | \( 1 + 28618 T^{2} - 68513757 T^{4} + 28618 p^{6} T^{6} + p^{12} T^{8} \) |
| 37 | $C_2^2$ | \( ( 1 - 254 T + 13863 T^{2} - 254 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 129392 T^{2} + p^{6} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 122 T + p^{3} T^{2} )^{4} \) |
| 47 | $C_2^3$ | \( 1 - 112598 T^{2} + 1899094275 T^{4} - 112598 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 170 T - 119977 T^{2} - 170 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 318308 T^{2} + 59139449223 T^{4} - 318308 p^{6} T^{6} + p^{12} T^{8} \) |
| 61 | $C_2^3$ | \( 1 - 84162 T^{2} - 44437132117 T^{4} - 84162 p^{6} T^{6} + p^{12} T^{8} \) |
| 67 | $C_2^2$ | \( ( 1 + 420 T - 124363 T^{2} + 420 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 420 T + p^{3} T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 - 116784 T^{2} - 137695723633 T^{4} - 116784 p^{6} T^{6} + p^{12} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 + 1052 T + 613665 T^{2} + 1052 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 957676 T^{2} + p^{6} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 - 358688 T^{2} - 368324209617 T^{4} - 358688 p^{6} T^{6} + p^{12} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 + 1725888 T^{2} + p^{6} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.862767247832296352430145649535, −8.246977845949687189804821289515, −8.211765970407081288472285208385, −7.87288930223953349072233701950, −7.36128598083077119459557177371, −7.19765426592738582970851080677, −6.92316564083665027488581113513, −6.87078866379999798331622116763, −6.84342087049541300895825129413, −5.99094697583361344284901146981, −5.96152296263960119714877878705, −5.53148148820618349535928801955, −5.32421845601274334500883835623, −4.80564077747380127918147423497, −4.35066009465779399305629374815, −4.30906963391460080816002741717, −4.19501202175393096788267810978, −3.45294991941480486610911878615, −3.37748447303658626694813475892, −2.75271583341484787248432384970, −2.45536795784349315405040783606, −1.71860391403002857647005742477, −1.40626738134325801287575532062, −0.986220153178945210003221434312, −0.65099415621894514482159646636,
0.65099415621894514482159646636, 0.986220153178945210003221434312, 1.40626738134325801287575532062, 1.71860391403002857647005742477, 2.45536795784349315405040783606, 2.75271583341484787248432384970, 3.37748447303658626694813475892, 3.45294991941480486610911878615, 4.19501202175393096788267810978, 4.30906963391460080816002741717, 4.35066009465779399305629374815, 4.80564077747380127918147423497, 5.32421845601274334500883835623, 5.53148148820618349535928801955, 5.96152296263960119714877878705, 5.99094697583361344284901146981, 6.84342087049541300895825129413, 6.87078866379999798331622116763, 6.92316564083665027488581113513, 7.19765426592738582970851080677, 7.36128598083077119459557177371, 7.87288930223953349072233701950, 8.211765970407081288472285208385, 8.246977845949687189804821289515, 8.862767247832296352430145649535
