| L(s) = 1 | − 36·9-s + 240·17-s + 328·25-s − 816·41-s − 536·49-s − 1.93e3·73-s + 810·81-s + 7.34e3·89-s − 2.96e3·97-s − 7.92e3·113-s + 8.02e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 8.64e3·153-s + 157-s + 163-s + 167-s + 296·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 4/3·9-s + 3.42·17-s + 2.62·25-s − 3.10·41-s − 1.56·49-s − 3.10·73-s + 10/9·81-s + 8.74·89-s − 3.09·97-s − 6.59·113-s + 6.02·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s − 4.56·153-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 0.134·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.113187903\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.113187903\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + p^{2} T^{2} )^{4} \) |
| good | 5 | \( ( 1 - 164 T^{2} + 19542 T^{4} - 164 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 7 | \( ( 1 + 268 T^{2} - 41658 T^{4} + 268 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 11 | \( ( 1 - 4012 T^{2} + 7272246 T^{4} - 4012 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 13 | \( ( 1 - 148 T^{2} + 3687126 T^{4} - 148 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 17 | \( ( 1 - 60 T + 6118 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
| 19 | \( ( 1 - 17932 T^{2} + 171826710 T^{4} - 17932 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 23 | \( ( 1 - 4900 T^{2} + 206522790 T^{4} - 4900 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 29 | \( ( 1 - 56516 T^{2} + 1986668214 T^{4} - 56516 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 31 | \( ( 1 + 63916 T^{2} + 2595558 p^{2} T^{4} + 63916 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 37 | \( ( 1 - 95476 T^{2} + 7028163510 T^{4} - 95476 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 41 | \( ( 1 + 204 T + 806 p T^{2} + 204 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
| 43 | \( ( 1 - 273964 T^{2} + 31085635254 T^{4} - 273964 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 47 | \( ( 1 + 112892 T^{2} + 23215757766 T^{4} + 112892 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 53 | \( ( 1 - 507620 T^{2} + 107623720470 T^{4} - 507620 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 59 | \( ( 1 - 121324 T^{2} - 26790709386 T^{4} - 121324 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 61 | \( ( 1 + 132332 T^{2} + 107394800406 T^{4} + 132332 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 67 | \( ( 1 - 364108 T^{2} + 112096368726 T^{4} - 364108 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 71 | \( ( 1 + 4612 p T^{2} + 275267100390 T^{4} + 4612 p^{7} T^{6} + p^{12} T^{8} )^{2} \) |
| 73 | \( ( 1 + 484 T + 541686 T^{2} + 484 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
| 79 | \( ( 1 + 932332 T^{2} + 435081520806 T^{4} + 932332 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 83 | \( ( 1 - 539404 T^{2} + 296977663254 T^{4} - 539404 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 89 | \( ( 1 - 1836 T + 2086774 T^{2} - 1836 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
| 97 | \( ( 1 + 740 T + 1943814 T^{2} + 740 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
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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
−4.61374689467733315313325985410, −4.30149400629757674648412390117, −4.28475167253485057076311849914, −4.16211174296075620475750147891, −3.83260629071676326944981521178, −3.76922321981611224359324571795, −3.46142582503955074894277109668, −3.39766114554702844661397290257, −3.28186833246346149849886744446, −3.14988076913166215555801434426, −3.00951455472590295018882837129, −2.86935578336640721240699483417, −2.83048369814012209256059605780, −2.70273725952707378567414619786, −2.07782175514304555586530936971, −2.03917448723456304114599333653, −2.03346861200024048402022574257, −1.64643899044992633638583610930, −1.50427433505096264048914611877, −1.34427720714433291050040697241, −0.942284741161076249285924439161, −0.853435154646252357519252737931, −0.72159107310488193175810629937, −0.44473068048442218055234134422, −0.07742737098673125996789373351,
0.07742737098673125996789373351, 0.44473068048442218055234134422, 0.72159107310488193175810629937, 0.853435154646252357519252737931, 0.942284741161076249285924439161, 1.34427720714433291050040697241, 1.50427433505096264048914611877, 1.64643899044992633638583610930, 2.03346861200024048402022574257, 2.03917448723456304114599333653, 2.07782175514304555586530936971, 2.70273725952707378567414619786, 2.83048369814012209256059605780, 2.86935578336640721240699483417, 3.00951455472590295018882837129, 3.14988076913166215555801434426, 3.28186833246346149849886744446, 3.39766114554702844661397290257, 3.46142582503955074894277109668, 3.76922321981611224359324571795, 3.83260629071676326944981521178, 4.16211174296075620475750147891, 4.28475167253485057076311849914, 4.30149400629757674648412390117, 4.61374689467733315313325985410
Plot not available for L-functions of degree greater than 10.
