| L(s) = 1 | − 4·4-s + 20·9-s + 2·16-s − 24·19-s + 272·31-s − 80·36-s − 28·49-s − 312·61-s − 36·64-s + 96·76-s − 256·79-s + 166·81-s − 512·109-s + 856·121-s − 1.08e3·124-s + 127-s + 131-s + 137-s + 139-s + 40·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.00e3·169-s − 480·171-s + ⋯ |
| L(s) = 1 | − 4-s + 20/9·9-s + 1/8·16-s − 1.26·19-s + 8.77·31-s − 2.22·36-s − 4/7·49-s − 5.11·61-s − 0.562·64-s + 1.26·76-s − 3.24·79-s + 2.04·81-s − 4.69·109-s + 7.07·121-s − 8.77·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 5/18·144-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 5.91·169-s − 2.80·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(6.386251722\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.386251722\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 20 T^{2} + 26 p^{2} T^{4} - 20 p^{4} T^{6} + p^{8} T^{8} \) |
| 5 | \( 1 \) |
| 7 | \( ( 1 + p T^{2} )^{4} \) |
| good | 2 | \( ( 1 + p T^{2} + 5 T^{4} + p^{5} T^{6} + p^{8} T^{8} )^{2} \) |
| 11 | \( ( 1 - 428 T^{2} + 74630 T^{4} - 428 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 13 | \( ( 1 - 500 T^{2} + 117354 T^{4} - 500 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 17 | \( ( 1 + 988 T^{2} + 407046 T^{4} + 988 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 19 | \( ( 1 + 6 T + 556 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 23 | \( ( 1 + 1444 T^{2} + 980166 T^{4} + 1444 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 29 | \( ( 1 - 2972 T^{2} + 3611558 T^{4} - 2972 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 31 | \( ( 1 - 68 T + 3050 T^{2} - 68 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 37 | \( ( 1 - 2700 T^{2} + 5483014 T^{4} - 2700 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 41 | \( ( 1 - 1292 T^{2} + 2832038 T^{4} - 1292 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 43 | \( ( 1 - 3692 T^{2} + 8632518 T^{4} - 3692 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 47 | \( ( 1 + 6148 T^{2} + 19144326 T^{4} + 6148 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 53 | \( ( 1 - 20 T^{2} - 13350138 T^{4} - 20 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 59 | \( ( 1 - 3676 T^{2} + 15964266 T^{4} - 3676 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 61 | \( ( 1 + 78 T + 8620 T^{2} + 78 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 67 | \( ( 1 - 16988 T^{2} + 112385766 T^{4} - 16988 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 71 | \( ( 1 - 10588 T^{2} + 78813510 T^{4} - 10588 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 73 | \( ( 1 - 11724 T^{2} + 89948134 T^{4} - 11724 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 79 | \( ( 1 + 64 T + 4434 T^{2} + 64 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 83 | \( ( 1 + 13948 T^{2} + 141899946 T^{4} + 13948 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 89 | \( ( 1 - 11468 T^{2} + 120945830 T^{4} - 11468 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 97 | \( ( 1 - 36732 T^{2} + 514361350 T^{4} - 36732 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
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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
−4.52779986446550508201064997650, −4.47797966368383683688004744060, −4.34614190532121129123610430305, −4.14567559639081434435758691615, −4.04888818447107646852848341439, −4.03165429706993927815201551912, −3.91191534894362798689482651886, −3.30896058108153887011519338116, −3.17327799657007449077573910942, −3.09363940805499963968038577497, −3.08157218231671675833505204568, −3.00389501577775449227561414677, −2.82933241469229204026137297807, −2.66880784293618619556696708474, −2.35594460415009739122921186981, −1.99517781029072979022218345859, −1.96660470912345887567370175019, −1.91533682047349905411500285524, −1.49430658549953275189948256997, −1.28365855846023892076999212243, −1.26644693876941901087365725537, −0.866848663763814196456229297777, −0.78599318206106636357645982164, −0.51384961675670253655508881327, −0.23989718834349543539577315248,
0.23989718834349543539577315248, 0.51384961675670253655508881327, 0.78599318206106636357645982164, 0.866848663763814196456229297777, 1.26644693876941901087365725537, 1.28365855846023892076999212243, 1.49430658549953275189948256997, 1.91533682047349905411500285524, 1.96660470912345887567370175019, 1.99517781029072979022218345859, 2.35594460415009739122921186981, 2.66880784293618619556696708474, 2.82933241469229204026137297807, 3.00389501577775449227561414677, 3.08157218231671675833505204568, 3.09363940805499963968038577497, 3.17327799657007449077573910942, 3.30896058108153887011519338116, 3.91191534894362798689482651886, 4.03165429706993927815201551912, 4.04888818447107646852848341439, 4.14567559639081434435758691615, 4.34614190532121129123610430305, 4.47797966368383683688004744060, 4.52779986446550508201064997650
Plot not available for L-functions of degree greater than 10.
