| L(s) = 1 | − 2·3-s − 3·4-s + 9-s + 6·12-s + 4·16-s + 12·17-s − 12·23-s − 15·25-s − 4·27-s − 18·29-s − 3·36-s − 10·43-s − 8·48-s − 24·51-s − 12·53-s + 4·61-s − 9·64-s − 36·68-s + 24·69-s + 30·75-s − 24·79-s − 4·81-s + 36·87-s + 36·92-s + 45·100-s + 30·101-s − 30·107-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 3/2·4-s + 1/3·9-s + 1.73·12-s + 16-s + 2.91·17-s − 2.50·23-s − 3·25-s − 0.769·27-s − 3.34·29-s − 1/2·36-s − 1.52·43-s − 1.15·48-s − 3.36·51-s − 1.64·53-s + 0.512·61-s − 9/8·64-s − 4.36·68-s + 2.88·69-s + 3.46·75-s − 2.70·79-s − 4/9·81-s + 3.85·87-s + 3.75·92-s + 9/2·100-s + 2.98·101-s − 2.90·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2^3$ | \( 1 + 3 T^{2} + 5 T^{4} + 3 p^{2} T^{6} + p^{4} T^{8} \) |
| 3 | $D_{4}$ | \( ( 1 + T + T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 5 | $D_4\times C_2$ | \( 1 + 3 p T^{2} + 101 T^{4} + 3 p^{3} T^{6} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 27 T^{2} + 377 T^{4} + 27 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{4} \) |
| 19 | $D_4\times C_2$ | \( 1 + 31 T^{2} + 537 T^{4} + 31 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 + 9 T + 73 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 96 T^{2} + 4910 T^{4} + 96 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 5 T + 45 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 78 T^{2} + 4595 T^{4} + 78 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 6 T + 31 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 6 T^{2} + 5627 T^{4} + 6 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 2 T - 66 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 118 T^{2} + 9435 T^{4} + 118 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 264 T^{2} + 27422 T^{4} + 264 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 + 134 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 83 | $D_4\times C_2$ | \( 1 + 270 T^{2} + 31667 T^{4} + 270 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 87 T^{2} + 1853 T^{4} + 87 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 103 T^{2} + 13485 T^{4} + 103 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−5.82283965533704249864137433334, −5.71570223720788005835521735337, −5.51948046883494235823935566341, −5.43408824262829632141378971710, −5.22648955931779422700274267481, −4.90976747342396851652808757254, −4.77173095065951897202247392764, −4.63917240028289197956813971939, −4.23852992229881899054544169637, −4.09987537926902573749292591649, −4.04806462246680120676345033995, −3.98795208474836277896977023973, −3.48048848693804697474930233579, −3.43463228160997762014783205089, −3.36950479475311550506925779534, −3.33579453188552984446874310907, −2.78690730538629641564743910182, −2.58871452474296397847341028630, −2.12184621440783039473598579230, −1.95129562410777885641795288495, −1.81787943167758195094476632093, −1.68368053573914757319938666775, −1.42370763263260325839201145816, −1.01324135603874719641719468789, −0.791226229635397120914887217480, 0, 0, 0, 0,
0.791226229635397120914887217480, 1.01324135603874719641719468789, 1.42370763263260325839201145816, 1.68368053573914757319938666775, 1.81787943167758195094476632093, 1.95129562410777885641795288495, 2.12184621440783039473598579230, 2.58871452474296397847341028630, 2.78690730538629641564743910182, 3.33579453188552984446874310907, 3.36950479475311550506925779534, 3.43463228160997762014783205089, 3.48048848693804697474930233579, 3.98795208474836277896977023973, 4.04806462246680120676345033995, 4.09987537926902573749292591649, 4.23852992229881899054544169637, 4.63917240028289197956813971939, 4.77173095065951897202247392764, 4.90976747342396851652808757254, 5.22648955931779422700274267481, 5.43408824262829632141378971710, 5.51948046883494235823935566341, 5.71570223720788005835521735337, 5.82283965533704249864137433334
