| L(s) = 1 | − 3·2-s − 3·3-s + 5·4-s − 6·5-s + 9·6-s − 6·8-s + 7·9-s + 18·10-s + 3·11-s − 15·12-s + 10·13-s + 18·15-s + 4·16-s + 6·17-s − 21·18-s + 3·19-s − 30·20-s − 9·22-s + 18·24-s + 5·25-s − 30·26-s − 18·27-s − 3·29-s − 54·30-s − 8·31-s − 9·33-s − 18·34-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 1.73·3-s + 5/2·4-s − 2.68·5-s + 3.67·6-s − 2.12·8-s + 7/3·9-s + 5.69·10-s + 0.904·11-s − 4.33·12-s + 2.77·13-s + 4.64·15-s + 16-s + 1.45·17-s − 4.94·18-s + 0.688·19-s − 6.70·20-s − 1.91·22-s + 3.67·24-s + 25-s − 5.88·26-s − 3.46·27-s − 0.557·29-s − 9.85·30-s − 1.43·31-s − 1.56·33-s − 3.08·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2264544832\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2264544832\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| good | 2 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - T^{2} + p^{2} T^{4} )( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} ) \) |
| 3 | $D_4\times C_2$ | \( 1 + p T + 2 T^{2} + p T^{3} + 13 T^{4} + p^{2} T^{5} + 2 p^{2} T^{6} + p^{4} T^{7} + p^{4} T^{8} \) |
| 5 | $D_{4}$ | \( ( 1 + 3 T + 11 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 3 T - 4 T^{2} + 27 T^{3} - 51 T^{4} + 27 p T^{5} - 4 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 6 T + 13 T^{2} + 66 T^{3} - 372 T^{4} + 66 p T^{5} + 13 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 3 T - 20 T^{2} + 27 T^{3} + 309 T^{4} + 27 p T^{5} - 20 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^3$ | \( 1 - 26 T^{2} + 147 T^{4} - 26 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 3 T - 20 T^{2} - 3 p T^{3} - 9 p T^{4} - 3 p^{2} T^{5} - 20 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 4 T + 21 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 4 T - 21 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 6 T - 50 T^{2} - 24 T^{3} + 4239 T^{4} - 24 p T^{5} - 50 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 5 T + 34 T^{2} - 475 T^{3} - 2843 T^{4} - 475 p T^{5} + 34 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 + 89 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 - 12 T + 137 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 113 T^{2} + 9288 T^{4} - 113 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 6 T - 25 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 12 T + 19 T^{2} + 108 T^{3} + 1488 T^{4} + 108 p T^{5} + 19 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_4\times C_2$ | \( 1 - 6 T + 10 T^{2} + 696 T^{3} - 6921 T^{4} + 696 p T^{5} + 10 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 121 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 21 T + 184 T^{2} - 1659 T^{3} + 18879 T^{4} - 1659 p T^{5} + 184 p^{2} T^{6} - 21 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 31 T + 538 T^{2} - 7099 T^{3} + 76303 T^{4} - 7099 p T^{5} + 538 p^{2} T^{6} - 31 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.66298446917750806771065471390, −7.28142430154969307186149356697, −7.25932534373386590949170932877, −7.20069662318010363832121629460, −6.99377040564688903640473413002, −6.45082244553320685183094117946, −6.12868206129805023025959856606, −6.00362683498816012288218274278, −5.94126395408244912607433308752, −5.70128077809935707480892536346, −5.30026066457416476673587814141, −4.82958074799276302840806630308, −4.78766583298122437668728241726, −4.32562077038803765419804744541, −3.84411349974681279307096852970, −3.69928828572747184626824010892, −3.67536858105666630378441493706, −3.47392850235825359097299150108, −3.36740292400398492670489980159, −2.15796666637193059853554129844, −1.96236745724617875761502886578, −1.66834253220644276771174324932, −0.953407325176680439080387772387, −0.846359738284369259111786564214, −0.39171063169947395409479960353,
0.39171063169947395409479960353, 0.846359738284369259111786564214, 0.953407325176680439080387772387, 1.66834253220644276771174324932, 1.96236745724617875761502886578, 2.15796666637193059853554129844, 3.36740292400398492670489980159, 3.47392850235825359097299150108, 3.67536858105666630378441493706, 3.69928828572747184626824010892, 3.84411349974681279307096852970, 4.32562077038803765419804744541, 4.78766583298122437668728241726, 4.82958074799276302840806630308, 5.30026066457416476673587814141, 5.70128077809935707480892536346, 5.94126395408244912607433308752, 6.00362683498816012288218274278, 6.12868206129805023025959856606, 6.45082244553320685183094117946, 6.99377040564688903640473413002, 7.20069662318010363832121629460, 7.25932534373386590949170932877, 7.28142430154969307186149356697, 7.66298446917750806771065471390
