| L(s) = 1 | − 28·3-s − 125·5-s + 104·7-s − 1.40e3·9-s + 5.14e3·11-s + 8.60e3·13-s + 3.50e3·15-s + 2.02e4·17-s − 4.55e4·19-s − 2.91e3·21-s − 7.20e4·23-s + 1.56e4·25-s + 1.00e5·27-s − 2.31e5·29-s − 8.01e4·31-s − 1.44e5·33-s − 1.30e4·35-s − 1.04e5·37-s − 2.40e5·39-s + 5.84e5·41-s + 7.95e5·43-s + 1.75e5·45-s + 4.25e5·47-s − 8.12e5·49-s − 5.67e5·51-s − 1.50e6·53-s − 6.43e5·55-s + ⋯ |
| L(s) = 1 | − 0.598·3-s − 0.447·5-s + 0.114·7-s − 0.641·9-s + 1.16·11-s + 1.08·13-s + 0.267·15-s + 1.00·17-s − 1.52·19-s − 0.0686·21-s − 1.23·23-s + 1/5·25-s + 0.982·27-s − 1.76·29-s − 0.483·31-s − 0.698·33-s − 0.0512·35-s − 0.339·37-s − 0.650·39-s + 1.32·41-s + 1.52·43-s + 0.286·45-s + 0.598·47-s − 0.986·49-s − 0.599·51-s − 1.38·53-s − 0.521·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.305989926\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.305989926\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + p^{3} T \) |
| good | 3 | \( 1 + 28 T + p^{7} T^{2} \) |
| 7 | \( 1 - 104 T + p^{7} T^{2} \) |
| 11 | \( 1 - 468 p T + p^{7} T^{2} \) |
| 13 | \( 1 - 8602 T + p^{7} T^{2} \) |
| 17 | \( 1 - 20274 T + p^{7} T^{2} \) |
| 19 | \( 1 + 45500 T + p^{7} T^{2} \) |
| 23 | \( 1 + 72072 T + p^{7} T^{2} \) |
| 29 | \( 1 + 231510 T + p^{7} T^{2} \) |
| 31 | \( 1 + 80128 T + p^{7} T^{2} \) |
| 37 | \( 1 + 104654 T + p^{7} T^{2} \) |
| 41 | \( 1 - 584922 T + p^{7} T^{2} \) |
| 43 | \( 1 - 795532 T + p^{7} T^{2} \) |
| 47 | \( 1 - 425664 T + p^{7} T^{2} \) |
| 53 | \( 1 + 1500798 T + p^{7} T^{2} \) |
| 59 | \( 1 + 246420 T + p^{7} T^{2} \) |
| 61 | \( 1 + 893942 T + p^{7} T^{2} \) |
| 67 | \( 1 - 2336836 T + p^{7} T^{2} \) |
| 71 | \( 1 + 203688 T + p^{7} T^{2} \) |
| 73 | \( 1 + 3805702 T + p^{7} T^{2} \) |
| 79 | \( 1 - 5053040 T + p^{7} T^{2} \) |
| 83 | \( 1 - 45492 T + p^{7} T^{2} \) |
| 89 | \( 1 - 980010 T + p^{7} T^{2} \) |
| 97 | \( 1 + 5247646 T + p^{7} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.78292826929728258784937754998, −9.402065428747945968913369925577, −8.555215945114071572914060581979, −7.60259046591724441187781718629, −6.26807426974674715419724349178, −5.79218102667946147482632775122, −4.29452867393941637575385775198, −3.49303131512706734708429746933, −1.82443044167310314420885783277, −0.57259036041916451671243368197,
0.57259036041916451671243368197, 1.82443044167310314420885783277, 3.49303131512706734708429746933, 4.29452867393941637575385775198, 5.79218102667946147482632775122, 6.26807426974674715419724349178, 7.60259046591724441187781718629, 8.555215945114071572914060581979, 9.402065428747945968913369925577, 10.78292826929728258784937754998
