| L(s) = 1 | − 28·9-s + 12·19-s + 112·31-s − 212·41-s + 294·49-s − 4·59-s − 28·71-s + 344·81-s − 556·101-s − 388·109-s + 444·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 124·169-s − 336·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 3.11·9-s + 0.631·19-s + 3.61·31-s − 5.17·41-s + 6·49-s − 0.0677·59-s − 0.394·71-s + 4.24·81-s − 5.50·101-s − 3.55·109-s + 3.66·121-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 + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 0.733·169-s − 1.96·171-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{16} \cdot 31^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{16} \cdot 31^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(4.560219360\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.560219360\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 31 | \( ( 1 - 56 T + 2350 T^{2} - 56 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| good | 3 | \( ( 1 + 14 T^{2} + 122 T^{4} + 14 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 7 | \( ( 1 - 3 p^{2} T^{2} + 10004 T^{4} - 3 p^{6} T^{6} + p^{8} T^{8} )^{2} \) |
| 11 | \( ( 1 - 222 T^{2} + 37242 T^{4} - 222 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 13 | \( ( 1 + 62 T^{2} + 53722 T^{4} + 62 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 17 | \( ( 1 - 300 T^{2} + 183846 T^{4} - 300 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 19 | \( ( 1 - 3 T + 524 T^{2} - 3 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 23 | \( ( 1 + 1532 T^{2} + 1076662 T^{4} + 1532 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 29 | \( ( 1 - 686 T^{2} + 1506490 T^{4} - 686 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 37 | \( ( 1 + 4270 T^{2} + 7953306 T^{4} + 4270 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 41 | \( ( 1 + 53 T + 2974 T^{2} + 53 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 43 | \( ( 1 + 7374 T^{2} + 20431482 T^{4} + 7374 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 47 | \( ( 1 - 3136 T^{2} + 6655486 T^{4} - 3136 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 53 | \( ( 1 + 5150 T^{2} + 18220666 T^{4} + 5150 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 59 | \( ( 1 + T + 5160 T^{2} + p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 61 | \( ( 1 - 5646 T^{2} + 29782650 T^{4} - 5646 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 67 | \( ( 1 - 9052 T^{2} + 54954214 T^{4} - 9052 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 71 | \( ( 1 + 7 T + 5088 T^{2} + 7 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 73 | \( ( 1 + 18924 T^{2} + 144957462 T^{4} + 18924 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 79 | \( ( 1 - 2036 T^{2} - 50938010 T^{4} - 2036 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 83 | \( ( 1 + 9134 T^{2} + 113467162 T^{4} + 9134 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 89 | \( ( 1 - 8892 T^{2} + 142105782 T^{4} - 8892 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 97 | \( ( 1 - 26995 T^{2} + 335773348 T^{4} - 26995 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
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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
−3.24309819991785164406510461435, −3.10765682812244210504681708710, −3.05480961293234939631424469450, −3.01762587586177678669170387317, −3.00689472780916102795359392364, −2.86973091161757514196314818362, −2.68608370115570421500660135541, −2.64297221713433702099927659281, −2.50792035515553413323711963596, −2.41480415500772770973059641877, −2.23962906982412321080488698892, −2.21332035611078013148038741946, −1.89537532635625249062313621524, −1.85716731491148182830419221881, −1.54941672313410226361432843040, −1.53107604900801372090590234522, −1.36623993949154566827065520743, −1.22049720249565663410396347163, −1.16660288365276373740030162196, −0.806791706872054388452594391496, −0.73624084493687439691737477913, −0.51555149643290822178761592891, −0.41451300848603453488783996075, −0.31396764679436622902269122789, −0.15555795276572203446673936831,
0.15555795276572203446673936831, 0.31396764679436622902269122789, 0.41451300848603453488783996075, 0.51555149643290822178761592891, 0.73624084493687439691737477913, 0.806791706872054388452594391496, 1.16660288365276373740030162196, 1.22049720249565663410396347163, 1.36623993949154566827065520743, 1.53107604900801372090590234522, 1.54941672313410226361432843040, 1.85716731491148182830419221881, 1.89537532635625249062313621524, 2.21332035611078013148038741946, 2.23962906982412321080488698892, 2.41480415500772770973059641877, 2.50792035515553413323711963596, 2.64297221713433702099927659281, 2.68608370115570421500660135541, 2.86973091161757514196314818362, 3.00689472780916102795359392364, 3.01762587586177678669170387317, 3.05480961293234939631424469450, 3.10765682812244210504681708710, 3.24309819991785164406510461435
Plot not available for L-functions of degree greater than 10.
