| L(s) = 1 | + 5·2-s + 18·3-s − 4-s + 90·6-s + 286·7-s + 25·8-s + 243·9-s − 242·11-s − 18·12-s + 166·13-s + 1.43e3·14-s − 73·16-s + 800·17-s + 1.21e3·18-s − 1.47e3·19-s + 5.14e3·21-s − 1.21e3·22-s + 3.37e3·23-s + 450·24-s + 830·26-s + 2.91e3·27-s − 286·28-s + 6.60e3·29-s − 7.52e3·31-s − 6.29e3·32-s − 4.35e3·33-s + 4.00e3·34-s + ⋯ |
| L(s) = 1 | + 0.883·2-s + 1.15·3-s − 0.0312·4-s + 1.02·6-s + 2.20·7-s + 0.138·8-s + 9-s − 0.603·11-s − 0.0360·12-s + 0.272·13-s + 1.94·14-s − 0.0712·16-s + 0.671·17-s + 0.883·18-s − 0.937·19-s + 2.54·21-s − 0.533·22-s + 1.32·23-s + 0.159·24-s + 0.240·26-s + 0.769·27-s − 0.0689·28-s + 1.45·29-s − 1.40·31-s − 1.08·32-s − 0.696·33-s + 0.593·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 680625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680625 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(15.48919700\) |
| \(L(\frac12)\) |
\(\approx\) |
\(15.48919700\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 5 T + 13 p T^{2} - 5 p^{5} T^{3} + p^{10} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 286 T + 49638 T^{2} - 286 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 166 T + 685578 T^{2} - 166 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 800 T + 2840414 T^{2} - 800 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 1476 T + 5490470 T^{2} + 1476 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 3370 T + 9784358 T^{2} - 3370 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 6600 T + 44401126 T^{2} - 6600 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 7528 T + 66189630 T^{2} + 7528 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 29916 T + 361611230 T^{2} - 29916 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 5780 T + 217632230 T^{2} + 5780 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 16656 T + 264340262 T^{2} - 16656 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 7850 T + 191750726 T^{2} + 7850 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 14178 T + 671305114 T^{2} + 14178 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 17300 T + 1408269110 T^{2} - 17300 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 2946 T + 1451133506 T^{2} + 2946 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 31336 T + 2492616438 T^{2} + 31336 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 33810 T + 3469543750 T^{2} + 33810 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 60644 T + 2552552022 T^{2} + 60644 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 1870 T + 2176543686 T^{2} - 1870 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 58296 T + 101571026 p T^{2} - 58296 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 92388 T + 3271275766 T^{2} - 92388 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 7120 T - 2888428386 T^{2} + 7120 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−9.399472366885101029866247140743, −9.345559203837308024247962038886, −8.638413156772128477277174772191, −8.403279613105015977521709977681, −7.943752124239055933766115873276, −7.68323922743350348139970946069, −7.34787180078595464951663735676, −6.74529481194748802660006530881, −6.05133462678771842134318129445, −5.59280426523777388045727780825, −4.93930270400051264063442955535, −4.78691767453622707349105321805, −4.20846047100684435872421812649, −4.16980003219276927072116143095, −3.20498895299337076467000052134, −2.81115898687339793477373671002, −2.22773869420500529429195118287, −1.70782920590782774827014020203, −1.21608535605291894008587321271, −0.63686080834752111571620882207,
0.63686080834752111571620882207, 1.21608535605291894008587321271, 1.70782920590782774827014020203, 2.22773869420500529429195118287, 2.81115898687339793477373671002, 3.20498895299337076467000052134, 4.16980003219276927072116143095, 4.20846047100684435872421812649, 4.78691767453622707349105321805, 4.93930270400051264063442955535, 5.59280426523777388045727780825, 6.05133462678771842134318129445, 6.74529481194748802660006530881, 7.34787180078595464951663735676, 7.68323922743350348139970946069, 7.943752124239055933766115873276, 8.403279613105015977521709977681, 8.638413156772128477277174772191, 9.345559203837308024247962038886, 9.399472366885101029866247140743
