| L(s) = 1 | − 2·2-s − 24·4-s + 38·5-s + 40·8-s − 76·10-s − 424·11-s − 924·13-s − 304·16-s + 2.34e3·17-s − 360·19-s − 912·20-s + 848·22-s + 12·23-s − 1.46e3·25-s + 1.84e3·26-s + 7.05e3·29-s + 3.54e3·31-s + 3.07e3·32-s − 4.69e3·34-s + 1.10e4·37-s + 720·38-s + 1.52e3·40-s − 3.50e3·41-s − 1.26e4·43-s + 1.01e4·44-s − 24·46-s + 2.29e4·47-s + ⋯ |
| L(s) = 1 | − 0.353·2-s − 3/4·4-s + 0.679·5-s + 0.220·8-s − 0.240·10-s − 1.05·11-s − 1.51·13-s − 0.296·16-s + 1.96·17-s − 0.228·19-s − 0.509·20-s + 0.373·22-s + 0.00473·23-s − 0.469·25-s + 0.536·26-s + 1.55·29-s + 0.663·31-s + 0.530·32-s − 0.696·34-s + 1.33·37-s + 0.0808·38-s + 0.150·40-s − 0.325·41-s − 1.04·43-s + 0.792·44-s − 0.00167·46-s + 1.51·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.227438628\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.227438628\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + p T + 7 p^{2} T^{2} + p^{6} T^{3} + p^{10} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 38 T + 2911 T^{2} - 38 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 424 T + 347473 T^{2} + 424 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 924 T + 927022 T^{2} + 924 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 138 p T + 3570955 T^{2} - 138 p^{6} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 360 T + 2145625 T^{2} + 360 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 12 T + 12696565 T^{2} - 12 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 7052 T + 35324974 T^{2} - 7052 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 3548 T + 41490053 T^{2} - 3548 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 11090 T + 146343239 T^{2} - 11090 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 3500 T + 206898214 T^{2} + 3500 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 12680 T + 267378054 T^{2} + 12680 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 22956 T + 491525173 T^{2} - 22956 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 3042 T + 716414839 T^{2} + 3042 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 65808 T + 2502089257 T^{2} - 65808 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 42486 T + 1501996159 T^{2} + 42486 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 42312 T + 3116577793 T^{2} - 42312 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 2208 T + 3433192846 T^{2} - 2208 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 50506 T + 2773040987 T^{2} + 50506 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 9004 T + 5176592589 T^{2} - 9004 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 104328 T + 9837878230 T^{2} - 104328 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 26666 T + 8107102555 T^{2} - 26666 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 2156 p T + 28107307478 T^{2} - 2156 p^{6} T^{3} + p^{10} T^{4} \) |
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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
−10.28889319015732535972350739935, −10.11207979284970771425373509404, −9.539190092735992800108223732176, −9.513754762161066758765877049684, −8.684564311649365112105163975886, −8.316608106660271744799049174614, −7.74386408448820548330290241217, −7.61921877682417396642082379618, −6.83250732722577465365392309973, −6.35743810614738602675630156248, −5.64423790774398678481654016157, −5.35812864255038493127223488734, −4.78936294314943478534029878934, −4.45467313845485669988795822377, −3.61484826467904793548486344018, −2.84530991157356787320772025739, −2.50824676302521373147626318055, −1.82013590159465964429512760179, −0.73077884214685122394879259156, −0.57519829511452390250946893609,
0.57519829511452390250946893609, 0.73077884214685122394879259156, 1.82013590159465964429512760179, 2.50824676302521373147626318055, 2.84530991157356787320772025739, 3.61484826467904793548486344018, 4.45467313845485669988795822377, 4.78936294314943478534029878934, 5.35812864255038493127223488734, 5.64423790774398678481654016157, 6.35743810614738602675630156248, 6.83250732722577465365392309973, 7.61921877682417396642082379618, 7.74386408448820548330290241217, 8.316608106660271744799049174614, 8.684564311649365112105163975886, 9.513754762161066758765877049684, 9.539190092735992800108223732176, 10.11207979284970771425373509404, 10.28889319015732535972350739935
