| L(s) = 1 | + 8·2-s − 82·3-s + 448·5-s − 656·6-s − 512·8-s + 2.18e3·9-s + 3.58e3·10-s − 2.40e3·11-s − 1.42e4·13-s − 3.67e4·15-s − 4.09e3·16-s + 2.48e3·17-s + 1.74e4·18-s + 3.64e4·19-s − 1.92e4·22-s + 1.28e4·23-s + 4.19e4·24-s + 7.81e4·25-s − 1.13e5·26-s + 1.33e4·27-s − 1.76e5·29-s − 2.93e5·30-s + 2.82e5·31-s + 1.97e5·33-s + 1.98e4·34-s + 2.14e5·37-s + 2.91e5·38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.75·3-s + 1.60·5-s − 1.23·6-s − 0.353·8-s + 9-s + 1.13·10-s − 0.545·11-s − 1.79·13-s − 2.81·15-s − 1/4·16-s + 0.122·17-s + 0.707·18-s + 1.22·19-s − 0.385·22-s + 0.220·23-s + 0.619·24-s + 25-s − 1.27·26-s + 0.130·27-s − 1.34·29-s − 1.98·30-s + 1.70·31-s + 0.956·33-s + 0.0867·34-s + 0.696·37-s + 0.862·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.334390291\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.334390291\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_2$ | \( 1 - p^{3} T + p^{6} T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 82 T + 4537 T^{2} + 82 p^{7} T^{3} + p^{14} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 448 T + 122579 T^{2} - 448 p^{7} T^{3} + p^{14} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2408 T - 13688707 T^{2} + 2408 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 7116 T + p^{7} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 2486 T - 404158477 T^{2} - 2486 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 36482 T + 437064585 T^{2} - 36482 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 560 p T - 6122743 p^{2} T^{2} - 560 p^{8} T^{3} + p^{14} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 88094 T + p^{7} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 282636 T + 52370494385 T^{2} - 282636 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 214534 T - 48907039977 T^{2} - 214534 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 140874 T + p^{7} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 848 p T + p^{7} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 716868 T + 7276608961 T^{2} - 716868 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 56946 T - 1171468292921 T^{2} - 56946 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 2149862 T + 2133255134225 T^{2} + 2149862 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 3084360 T + 6370533773579 T^{2} - 3084360 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 3034364 T + 3146653279173 T^{2} - 3034364 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 106624 T + p^{7} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 988930 T - 10069415974197 T^{2} - 988930 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 3415896 T - 7535563503343 T^{2} + 3415896 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 15142 T + p^{7} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 174810 T - 44200776359429 T^{2} - 174810 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 13506790 T + p^{7} T^{2} )^{2} \) |
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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
−13.71720789442631473406257196799, −12.09654555222135561344101433674, −12.05511929407332950370009917858, −11.04556966892293018148239114911, −10.99152858455901272443890732296, −10.02144975301279993407640641713, −9.580000511810393724058574099492, −9.548297436095923527476968886677, −8.315672704508930245738036361036, −7.48817010044403204893241079587, −6.88360500472537270248354219096, −6.19612510686472654055054516714, −5.58610327581473612632563744507, −5.42458837748998954429958543458, −4.96854211467432074996184161531, −4.15986329505504197080669206365, −2.79634909318588009037044448766, −2.39674511386797836119128447927, −1.23613844359551030167109484403, −0.38032429338712925123982072119,
0.38032429338712925123982072119, 1.23613844359551030167109484403, 2.39674511386797836119128447927, 2.79634909318588009037044448766, 4.15986329505504197080669206365, 4.96854211467432074996184161531, 5.42458837748998954429958543458, 5.58610327581473612632563744507, 6.19612510686472654055054516714, 6.88360500472537270248354219096, 7.48817010044403204893241079587, 8.315672704508930245738036361036, 9.548297436095923527476968886677, 9.580000511810393724058574099492, 10.02144975301279993407640641713, 10.99152858455901272443890732296, 11.04556966892293018148239114911, 12.05511929407332950370009917858, 12.09654555222135561344101433674, 13.71720789442631473406257196799
