| L(s) = 1 | + 54·3-s − 264·5-s + 2.18e3·9-s − 4.98e3·11-s + 1.01e4·13-s − 1.42e4·15-s − 1.78e4·17-s − 6.25e3·19-s + 1.40e4·23-s + 2.73e4·25-s + 7.87e4·27-s + 2.43e5·29-s − 4.70e5·31-s − 2.68e5·33-s + 3.11e5·37-s + 5.47e5·39-s − 9.19e5·41-s − 1.12e5·43-s − 5.77e5·45-s + 1.02e5·47-s − 9.62e5·51-s + 2.72e6·53-s + 1.31e6·55-s − 3.37e5·57-s + 1.90e4·59-s + 9.25e5·61-s − 2.67e6·65-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.944·5-s + 9-s − 1.12·11-s + 1.28·13-s − 1.09·15-s − 0.880·17-s − 0.209·19-s + 0.240·23-s + 0.350·25-s + 0.769·27-s + 1.85·29-s − 2.83·31-s − 1.30·33-s + 1.00·37-s + 1.47·39-s − 2.08·41-s − 0.216·43-s − 0.944·45-s + 0.143·47-s − 1.01·51-s + 2.50·53-s + 1.06·55-s − 0.241·57-s + 0.0120·59-s + 0.521·61-s − 1.21·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - p^{3} T )^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 + 264 T + 8462 p T^{2} + 264 p^{7} T^{3} + p^{14} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4980 T + 38737606 T^{2} + 4980 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 10148 T + 75576846 T^{2} - 10148 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 17832 T + 345159502 T^{2} + 17832 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 6256 T + 1754965926 T^{2} + 6256 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 14052 T + 6233591566 T^{2} - 14052 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 243588 T + 49005121054 T^{2} - 243588 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 470824 T + 110401476030 T^{2} + 470824 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 311116 T + 134140188030 T^{2} - 311116 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 919248 T + 579002453902 T^{2} + 919248 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 112616 T + 296638211478 T^{2} + 112616 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 102456 T + 463813338910 T^{2} - 102456 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 2720028 T + 3726830950846 T^{2} - 2720028 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 19008 T - 1663332768746 T^{2} - 19008 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 925148 T + 2505443574174 T^{2} - 925148 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 2053424 T + 3053533142214 T^{2} + 2053424 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 869508 T + 14567098422382 T^{2} + 869508 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 3505228 T + 11289516695814 T^{2} + 3505228 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 6640856 T + 37612376152158 T^{2} + 6640856 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 7856760 T + 50010766932550 T^{2} + 7856760 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 9330384 T + 109971781045486 T^{2} - 9330384 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 2220212 T + 72228119801046 T^{2} - 2220212 p^{7} T^{3} + p^{14} T^{4} \) |
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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
−8.934971898332537283404370285011, −8.798662546833700358922201995876, −8.646411207577870492610944207758, −8.114020566985749368065014144628, −7.56885668382840854627352854006, −7.35972993514765062641935221386, −6.76758102189906083857653811949, −6.39626558153849916078332449177, −5.56302387563977546706794249061, −5.26953067631731195814860873648, −4.44262303808077935854672547105, −4.21866041241976576314938099481, −3.55391345850947470063644258513, −3.32705302034246304111447564702, −2.58206872851000599742630900540, −2.29047745092843433565209089328, −1.47197259788083612472288540204, −1.05772834885090134828798593353, 0, 0,
1.05772834885090134828798593353, 1.47197259788083612472288540204, 2.29047745092843433565209089328, 2.58206872851000599742630900540, 3.32705302034246304111447564702, 3.55391345850947470063644258513, 4.21866041241976576314938099481, 4.44262303808077935854672547105, 5.26953067631731195814860873648, 5.56302387563977546706794249061, 6.39626558153849916078332449177, 6.76758102189906083857653811949, 7.35972993514765062641935221386, 7.56885668382840854627352854006, 8.114020566985749368065014144628, 8.646411207577870492610944207758, 8.798662546833700358922201995876, 8.934971898332537283404370285011
