| L(s) = 1 | + 6·5-s − 14·7-s − 44·11-s + 14·13-s − 96·17-s + 170·19-s − 152·23-s − 186·25-s + 128·29-s + 68·31-s − 84·35-s − 256·37-s − 88·41-s + 724·43-s + 244·47-s + 147·49-s − 188·53-s − 264·55-s − 138·59-s − 358·61-s + 84·65-s + 200·67-s + 1.40e3·71-s − 628·73-s + 616·77-s + 200·79-s − 618·83-s + ⋯ |
| L(s) = 1 | + 0.536·5-s − 0.755·7-s − 1.20·11-s + 0.298·13-s − 1.36·17-s + 2.05·19-s − 1.37·23-s − 1.48·25-s + 0.819·29-s + 0.393·31-s − 0.405·35-s − 1.13·37-s − 0.335·41-s + 2.56·43-s + 0.757·47-s + 3/7·49-s − 0.487·53-s − 0.647·55-s − 0.304·59-s − 0.751·61-s + 0.160·65-s + 0.364·67-s + 2.34·71-s − 1.00·73-s + 0.911·77-s + 0.284·79-s − 0.817·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4064256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4064256 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.760603497\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.760603497\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 6 T + 222 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 p T + 2998 T^{2} + 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 14 T + 4406 T^{2} - 14 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 96 T + 8430 T^{2} + 96 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 170 T + 17946 T^{2} - 170 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 152 T + 20638 T^{2} + 152 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 128 T + 10102 T^{2} - 128 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 68 T + 35726 T^{2} - 68 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 256 T + 117542 T^{2} + 256 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 88 T + 15310 T^{2} + 88 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 724 T + 278070 T^{2} - 724 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 244 T + 204622 T^{2} - 244 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 188 T + 155038 T^{2} + 188 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 138 T + 375226 T^{2} + 138 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 358 T + 483006 T^{2} + 358 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 200 T - 155706 T^{2} - 200 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 1400 T + 1176814 T^{2} - 1400 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 628 T + 378758 T^{2} + 628 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 200 T + 926 T^{2} - 200 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 618 T + 303658 T^{2} + 618 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 908 T + 1379254 T^{2} - 908 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 1520 T + 2349518 T^{2} - 1520 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.880730200010781061170697447597, −8.832665021818482595420667918677, −7.972114605370291233008936705477, −7.966065927751159754290518805551, −7.30618466209904255432450898001, −7.26347006194653955618170671546, −6.35716123561414578856282981700, −6.34973763427755128948083396360, −5.76560284056879509570600110859, −5.53305298363128882336889889019, −5.03386720377941119858194666967, −4.56261060720678862331468032819, −3.97794739898769850186288261812, −3.68684628872805623913036710790, −2.86829158737264018419057918460, −2.83127763644771024212229708793, −1.94151559880769832902064533960, −1.87840204402515847678699805115, −0.70344676301803798341588254032, −0.47799133750834560919848247047,
0.47799133750834560919848247047, 0.70344676301803798341588254032, 1.87840204402515847678699805115, 1.94151559880769832902064533960, 2.83127763644771024212229708793, 2.86829158737264018419057918460, 3.68684628872805623913036710790, 3.97794739898769850186288261812, 4.56261060720678862331468032819, 5.03386720377941119858194666967, 5.53305298363128882336889889019, 5.76560284056879509570600110859, 6.34973763427755128948083396360, 6.35716123561414578856282981700, 7.26347006194653955618170671546, 7.30618466209904255432450898001, 7.966065927751159754290518805551, 7.972114605370291233008936705477, 8.832665021818482595420667918677, 8.880730200010781061170697447597
