L(s) = 1 | − 2·5-s + 17·9-s − 48·13-s − 2·17-s − 47·25-s + 48·29-s − 98·37-s + 96·41-s − 34·45-s − 50·53-s + 2·61-s + 96·65-s − 190·73-s + 208·81-s + 4·85-s − 190·89-s − 288·97-s − 146·101-s − 142·109-s + 192·113-s − 816·117-s − 47·121-s + 146·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
L(s) = 1 | − 2/5·5-s + 17/9·9-s − 3.69·13-s − 0.117·17-s − 1.87·25-s + 1.65·29-s − 2.64·37-s + 2.34·41-s − 0.755·45-s − 0.943·53-s + 2/61·61-s + 1.47·65-s − 2.60·73-s + 2.56·81-s + 4/85·85-s − 2.13·89-s − 2.96·97-s − 1.44·101-s − 1.30·109-s + 1.69·113-s − 6.97·117-s − 0.388·121-s + 1.16·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2458624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2458624 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.02037500047\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.02037500047\) |
\(L(2)\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
good | 3 | $C_2^2$ | \( 1 - 17 T^{2} + p^{4} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 47 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 24 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T + p^{2} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 673 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 1009 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 24 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 241 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 49 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 48 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 3122 T^{2} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 1393 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 25 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 6673 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 4753 T^{2} + p^{4} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 866 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 95 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 10801 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 8594 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 95 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 144 T + p^{2} T^{2} )^{2} \) |
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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
−9.585671589860434202168201371447, −9.208845359957239238337697286015, −8.684249048673170416666209227502, −7.973158524482111106585256454299, −7.82994512642282414238231101193, −7.26532360147701610936480437099, −7.24493519917293213811890555274, −6.83378794036345885091336655283, −6.39024359585340319699812337716, −5.44964170641966762550519063799, −5.44252710686740174706948520646, −4.56957747946361817481303712848, −4.55318899343814814947202738878, −4.18926273854097998308499131918, −3.53569745841225698389206515271, −2.67428628889902789518317161807, −2.52668993365403805641694438733, −1.79748825101306848895504491385, −1.26876705039339875343631158995, −0.03867552626891854107382691626,
0.03867552626891854107382691626, 1.26876705039339875343631158995, 1.79748825101306848895504491385, 2.52668993365403805641694438733, 2.67428628889902789518317161807, 3.53569745841225698389206515271, 4.18926273854097998308499131918, 4.55318899343814814947202738878, 4.56957747946361817481303712848, 5.44252710686740174706948520646, 5.44964170641966762550519063799, 6.39024359585340319699812337716, 6.83378794036345885091336655283, 7.24493519917293213811890555274, 7.26532360147701610936480437099, 7.82994512642282414238231101193, 7.973158524482111106585256454299, 8.684249048673170416666209227502, 9.208845359957239238337697286015, 9.585671589860434202168201371447