L(s) = 1 | − 32·9-s − 48·17-s − 20·25-s + 224·41-s + 192·49-s − 528·73-s + 356·81-s + 176·89-s − 528·97-s + 1.00e3·113-s − 168·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 1.53e3·153-s + 157-s + 163-s + 167-s + 1.20e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
L(s) = 1 | − 3.55·9-s − 2.82·17-s − 4/5·25-s + 5.46·41-s + 3.91·49-s − 7.23·73-s + 4.39·81-s + 1.97·89-s − 5.44·97-s + 8.92·113-s − 1.38·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 10.0·153-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 7.14·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2385828961\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2385828961\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( ( 1 + p T^{2} )^{4} \) |
good | 3 | \( ( 1 + 16 T^{2} + 206 T^{4} + 16 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 7 | \( ( 1 - 96 T^{2} + 6606 T^{4} - 96 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 11 | \( ( 1 + 84 T^{2} - 7674 T^{4} + 84 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 13 | \( ( 1 - 604 T^{2} + 147046 T^{4} - 604 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 17 | \( ( 1 + 12 T + 294 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 19 | \( ( 1 + 1124 T^{2} + 571366 T^{4} + 1124 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 23 | \( ( 1 - 1856 T^{2} + 1404046 T^{4} - 1856 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 29 | \( ( 1 - 3196 T^{2} + 3966886 T^{4} - 3196 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 31 | \( ( 1 - 1524 T^{2} + 1236966 T^{4} - 1524 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 37 | \( ( 1 - 4444 T^{2} + 8653606 T^{4} - 4444 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 41 | \( ( 1 - 56 T + 3966 T^{2} - 56 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 43 | \( ( 1 + 6896 T^{2} + 18665806 T^{4} + 6896 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 47 | \( ( 1 - 4736 T^{2} + 14725966 T^{4} - 4736 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 53 | \( ( 1 - 6364 T^{2} + 22034086 T^{4} - 6364 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 59 | \( ( 1 + 8164 T^{2} + 34261926 T^{4} + 8164 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 61 | \( ( 1 - 6076 T^{2} + 23076646 T^{4} - 6076 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 67 | \( ( 1 + 7536 T^{2} + 47276046 T^{4} + 7536 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 71 | \( ( 1 - 12084 T^{2} + 81146406 T^{4} - 12084 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 73 | \( ( 1 + 132 T + 9894 T^{2} + 132 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 79 | \( ( 1 - 11844 T^{2} + 72414726 T^{4} - 11844 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 83 | \( ( 1 + 21296 T^{2} + 201120526 T^{4} + 21296 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 89 | \( ( 1 - 44 T + 8326 T^{2} - 44 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 97 | \( ( 1 + 132 T + 22454 T^{2} + 132 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.03019963447455261436563577862, −3.90397800395657631795747517653, −3.44279817825870775667512526616, −3.39445576756201181337833047724, −3.35336970125939363198900325535, −3.27535412236647002132441948565, −3.03857918352473653928909796275, −2.84914290736239458010771211549, −2.80016687994969346067335790948, −2.61348083326064778858670369020, −2.58686208206637066030245333518, −2.55479733187826001259582787702, −2.45067407774836597713109783579, −2.04589752746125652668820769505, −1.93962637407056010237717449222, −1.90795510445816607268416886390, −1.90456168102444421342172914644, −1.52370738556090424097981385079, −1.20595767169820111504450443387, −0.995291120330281117625530598829, −0.880375176487838053912375316388, −0.67311497935157409497180207184, −0.41809923654717290356479194978, −0.35921341730030584926211462065, −0.05374533275680433171188999915,
0.05374533275680433171188999915, 0.35921341730030584926211462065, 0.41809923654717290356479194978, 0.67311497935157409497180207184, 0.880375176487838053912375316388, 0.995291120330281117625530598829, 1.20595767169820111504450443387, 1.52370738556090424097981385079, 1.90456168102444421342172914644, 1.90795510445816607268416886390, 1.93962637407056010237717449222, 2.04589752746125652668820769505, 2.45067407774836597713109783579, 2.55479733187826001259582787702, 2.58686208206637066030245333518, 2.61348083326064778858670369020, 2.80016687994969346067335790948, 2.84914290736239458010771211549, 3.03857918352473653928909796275, 3.27535412236647002132441948565, 3.35336970125939363198900325535, 3.39445576756201181337833047724, 3.44279817825870775667512526616, 3.90397800395657631795747517653, 4.03019963447455261436563577862
Plot not available for L-functions of degree greater than 10.