| L(s) = 1 | + 6·3-s + 6·5-s + 8·7-s + 27·9-s − 66·11-s − 2·13-s + 36·15-s − 34·17-s + 26·19-s + 48·21-s + 198·23-s + 65·25-s + 108·27-s + 444·29-s − 532·31-s − 396·33-s + 48·35-s + 88·37-s − 12·39-s + 570·41-s + 182·43-s + 162·45-s − 420·47-s − 350·49-s − 204·51-s − 300·53-s − 396·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.536·5-s + 0.431·7-s + 9-s − 1.80·11-s − 0.0426·13-s + 0.619·15-s − 0.485·17-s + 0.313·19-s + 0.498·21-s + 1.79·23-s + 0.519·25-s + 0.769·27-s + 2.84·29-s − 3.08·31-s − 2.08·33-s + 0.231·35-s + 0.391·37-s − 0.0492·39-s + 2.17·41-s + 0.645·43-s + 0.536·45-s − 1.30·47-s − 1.02·49-s − 0.560·51-s − 0.777·53-s − 0.970·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 665856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 665856 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.475923130\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.475923130\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 17 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 6 T - 29 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 8 T + 414 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 6 p T + 3463 T^{2} + 6 p^{4} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 3243 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 26 T + 9279 T^{2} - 26 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 198 T + 33847 T^{2} - 198 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 444 T + 93454 T^{2} - 444 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 532 T + 129186 T^{2} + 532 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 88 T + 29514 T^{2} - 88 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 570 T + 195739 T^{2} - 570 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 182 T + 162687 T^{2} - 182 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 420 T + 249154 T^{2} + 420 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 300 T + 43486 T^{2} + 300 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 444 T + 294154 T^{2} + 444 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 400 T + 55914 T^{2} - 400 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 968 T + 742470 T^{2} - 968 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 1848 T + 1513150 T^{2} - 1848 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 868 T + 963798 T^{2} - 868 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 254 T + p^{3} T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 12 p T + 1388986 T^{2} + 12 p^{4} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 288 T + 287602 T^{2} + 288 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 1024 T + 2038818 T^{2} - 1024 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.867524621875145021490405161122, −9.539530189297787365130959988724, −9.326575235248236713787547450084, −8.739430480933774683682437742660, −8.241935812489843689771078972203, −8.164718355711679463740583704887, −7.38884366348938379196404866598, −7.37288929107470924224065553371, −6.59691323272191981227127743911, −6.32012573332259333555707296740, −5.30913339743556374694414010919, −5.24024795078398242309622300718, −4.77636126636573459393378003116, −4.19365471440703557549245284298, −3.38273514131425903126774830768, −3.00730320549634443764432716870, −2.44495371971793750953314431637, −2.14136375086305265634130521952, −1.25786358721961424135900693925, −0.59541988677432637702061280445,
0.59541988677432637702061280445, 1.25786358721961424135900693925, 2.14136375086305265634130521952, 2.44495371971793750953314431637, 3.00730320549634443764432716870, 3.38273514131425903126774830768, 4.19365471440703557549245284298, 4.77636126636573459393378003116, 5.24024795078398242309622300718, 5.30913339743556374694414010919, 6.32012573332259333555707296740, 6.59691323272191981227127743911, 7.37288929107470924224065553371, 7.38884366348938379196404866598, 8.164718355711679463740583704887, 8.241935812489843689771078972203, 8.739430480933774683682437742660, 9.326575235248236713787547450084, 9.539530189297787365130959988724, 9.867524621875145021490405161122
