| L(s) = 1 | − 12·5-s − 44·7-s + 52·11-s − 26·13-s + 20·17-s + 60·19-s − 8·23-s + 30·25-s − 132·29-s + 140·31-s + 528·35-s + 68·37-s − 28·41-s + 36·47-s + 938·49-s − 668·53-s − 624·55-s − 508·59-s + 340·61-s + 312·65-s + 940·67-s + 300·71-s + 1.12e3·73-s − 2.28e3·77-s + 1.52e3·79-s + 524·83-s − 240·85-s + ⋯ |
| L(s) = 1 | − 1.07·5-s − 2.37·7-s + 1.42·11-s − 0.554·13-s + 0.285·17-s + 0.724·19-s − 0.0725·23-s + 6/25·25-s − 0.845·29-s + 0.811·31-s + 2.54·35-s + 0.302·37-s − 0.106·41-s + 0.111·47-s + 2.73·49-s − 1.73·53-s − 1.52·55-s − 1.12·59-s + 0.713·61-s + 0.595·65-s + 1.71·67-s + 0.501·71-s + 1.80·73-s − 3.38·77-s + 2.16·79-s + 0.692·83-s − 0.306·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3504384 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3504384 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 12 T + 114 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 44 T + 998 T^{2} + 44 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 26 T + p^{3} T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 20 T + 9238 T^{2} - 20 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 60 T + 10318 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 8 T - 418 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 132 T + 28366 T^{2} + 132 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 140 T + 25782 T^{2} - 140 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 68 T + 68750 T^{2} - 68 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 28 T + 129610 T^{2} + 28 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 2914 p T^{2} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 36 T + 201778 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 668 T + 406558 T^{2} + 668 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 508 T + 392026 T^{2} + 508 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 340 T + 471854 T^{2} - 340 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 940 T + 504398 T^{2} - 940 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 300 T - 57006 T^{2} - 300 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 1124 T + 1087686 T^{2} - 1124 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 1520 T + 1230686 T^{2} - 1520 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 524 T + 908810 T^{2} - 524 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 1900 T + 2312266 T^{2} + 1900 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 1436 T + 2285142 T^{2} + 1436 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.659314656201227123907407454359, −8.281070193100072734471404005118, −7.78710342068889981386247026438, −7.58492111603631264122147637485, −6.84131486050085617790809602983, −6.77124169218390644567232598628, −6.42864839786140833994736941152, −6.09929419067110008098845112542, −5.35483764183193332685489887111, −5.16319157197850804973390641030, −4.31791768881848020165286053694, −3.99440462185048265865660132166, −3.55414415699425161581413133450, −3.40417562057635315385649364014, −2.78038519231245257871463135579, −2.35075911058161871325755520422, −1.36369030630018271198301901855, −0.884414296797356043782937990923, 0, 0,
0.884414296797356043782937990923, 1.36369030630018271198301901855, 2.35075911058161871325755520422, 2.78038519231245257871463135579, 3.40417562057635315385649364014, 3.55414415699425161581413133450, 3.99440462185048265865660132166, 4.31791768881848020165286053694, 5.16319157197850804973390641030, 5.35483764183193332685489887111, 6.09929419067110008098845112542, 6.42864839786140833994736941152, 6.77124169218390644567232598628, 6.84131486050085617790809602983, 7.58492111603631264122147637485, 7.78710342068889981386247026438, 8.281070193100072734471404005118, 8.659314656201227123907407454359
