| L(s) = 1 | + 4·2-s + 12·4-s + 32·8-s + 28·11-s + 80·16-s + 112·22-s − 280·23-s + 142·25-s + 572·29-s + 192·32-s − 76·37-s − 68·43-s + 336·44-s − 1.12e3·46-s + 568·50-s + 148·53-s + 2.28e3·58-s + 448·64-s + 1.36e3·67-s − 1.17e3·71-s − 304·74-s + 2.44e3·79-s − 272·86-s + 896·88-s − 3.36e3·92-s + 1.70e3·100-s + 592·106-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 1.41·8-s + 0.767·11-s + 5/4·16-s + 1.08·22-s − 2.53·23-s + 1.13·25-s + 3.66·29-s + 1.06·32-s − 0.337·37-s − 0.241·43-s + 1.15·44-s − 3.58·46-s + 1.60·50-s + 0.383·53-s + 5.17·58-s + 7/8·64-s + 2.49·67-s − 1.96·71-s − 0.477·74-s + 3.47·79-s − 0.341·86-s + 1.08·88-s − 3.80·92-s + 1.70·100-s + 0.542·106-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(9.854919322\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.854919322\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 142 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 14 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 1802 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 9824 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 13716 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 140 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 286 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 50870 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 38 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 122000 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 34 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 66154 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 74 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 222260 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 453762 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 684 T + p^{3} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 588 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 705072 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 1220 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 964772 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 1028000 T^{2} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 375456 T^{2} + p^{6} T^{4} \) |
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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
−9.946544502531796978791203499092, −9.898343925728777443595474818229, −9.014929961991690208959117579016, −8.640081481245795414438356933812, −8.119705103718236217073820959508, −7.919657262481967479203518122031, −7.20133251311461425350983208211, −6.68241232975560193979364044125, −6.39083494036806901929692551802, −6.21924781091340677971233010124, −5.49922180824754197246436543132, −5.03989916796932359090620313456, −4.48845378918428818031632182992, −4.31074968961621518905078370274, −3.56379522214150126113651148778, −3.27655088289964467490530643500, −2.46627163507187380892450697437, −2.15437985690133539072925133592, −1.26920257380822685409214092550, −0.66752804351861086521769475915,
0.66752804351861086521769475915, 1.26920257380822685409214092550, 2.15437985690133539072925133592, 2.46627163507187380892450697437, 3.27655088289964467490530643500, 3.56379522214150126113651148778, 4.31074968961621518905078370274, 4.48845378918428818031632182992, 5.03989916796932359090620313456, 5.49922180824754197246436543132, 6.21924781091340677971233010124, 6.39083494036806901929692551802, 6.68241232975560193979364044125, 7.20133251311461425350983208211, 7.919657262481967479203518122031, 8.119705103718236217073820959508, 8.640081481245795414438356933812, 9.014929961991690208959117579016, 9.898343925728777443595474818229, 9.946544502531796978791203499092
