| L(s) = 1 | + 16·2-s − 70·3-s + 192·4-s − 126·5-s − 1.12e3·6-s + 2.04e3·8-s + 1.27e3·9-s − 2.01e3·10-s − 3.42e3·11-s − 1.34e4·12-s + 6.39e3·13-s + 8.82e3·15-s + 2.04e4·16-s + 3.84e4·17-s + 2.03e4·18-s + 4.33e4·19-s − 2.41e4·20-s − 5.47e4·22-s + 8.99e4·23-s − 1.43e5·24-s + 1.51e4·25-s + 1.02e5·26-s + 1.21e4·27-s + 1.59e5·29-s + 1.41e5·30-s + 1.43e5·31-s + 1.96e5·32-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.49·3-s + 3/2·4-s − 0.450·5-s − 2.11·6-s + 1.41·8-s + 0.580·9-s − 0.637·10-s − 0.774·11-s − 2.24·12-s + 0.807·13-s + 0.674·15-s + 5/4·16-s + 1.89·17-s + 0.821·18-s + 1.45·19-s − 0.676·20-s − 1.09·22-s + 1.54·23-s − 2.11·24-s + 0.193·25-s + 1.14·26-s + 0.118·27-s + 1.21·29-s + 0.954·30-s + 0.865·31-s + 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(5.312092107\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.312092107\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 - p^{3} T )^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 + 70 T + 1210 p T^{2} + 70 p^{7} T^{3} + p^{14} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 126 T + 146 p T^{2} + 126 p^{7} T^{3} + p^{14} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 3420 T + 10638598 T^{2} + 3420 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 6398 T + 65395986 T^{2} - 6398 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 38472 T + 1174752142 T^{2} - 38472 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2282 p T + 2141455038 T^{2} - 2282 p^{8} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 89928 T + 8706372814 T^{2} - 89928 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 159576 T + 29581073878 T^{2} - 159576 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 143612 T + 59486595774 T^{2} - 143612 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 271832 T + 32500290822 T^{2} + 271832 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 64848 T + 315504978094 T^{2} + 64848 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 1527964 T + 1127024379942 T^{2} - 1527964 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 485436 T + 473142464734 T^{2} + 485436 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 145716 T + 2270203949662 T^{2} + 145716 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 4183662 T + 9350646993118 T^{2} - 4183662 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 280658 T + 5246022775002 T^{2} - 280658 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 5671648 T + 17884478342022 T^{2} - 5671648 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 619272 T + 17339691732334 T^{2} + 619272 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 3939628 T + 25580837567814 T^{2} + 3939628 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 4656616 T + 21455665606878 T^{2} - 4656616 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 1235850 T + 18764123455390 T^{2} + 1235850 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 17241420 T + 151950390368758 T^{2} - 17241420 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 740936 T + 158749094330286 T^{2} - 740936 p^{7} T^{3} + p^{14} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.59059145512418500563991253843, −12.30631376923337532759570823072, −11.60549893827281314017899102130, −11.57802924788042977731720451668, −10.90759120553763303453884588948, −10.42049096863479230413302500666, −9.937910990469755705123788506379, −8.953598756688155337480425739233, −8.016049455537014194907331760595, −7.60395977146959051636679440587, −6.86404682627677443441957909176, −6.31322357990934694238916343782, −5.46224941466832462632207094884, −5.42203168498554176502115854830, −4.86646254159430504719123436781, −3.85660470159612266251612217368, −3.24494489675013068744717284309, −2.58398357477723331689440866672, −0.944126416178094810963707664444, −0.887553383997339799834834808940,
0.887553383997339799834834808940, 0.944126416178094810963707664444, 2.58398357477723331689440866672, 3.24494489675013068744717284309, 3.85660470159612266251612217368, 4.86646254159430504719123436781, 5.42203168498554176502115854830, 5.46224941466832462632207094884, 6.31322357990934694238916343782, 6.86404682627677443441957909176, 7.60395977146959051636679440587, 8.016049455537014194907331760595, 8.953598756688155337480425739233, 9.937910990469755705123788506379, 10.42049096863479230413302500666, 10.90759120553763303453884588948, 11.57802924788042977731720451668, 11.60549893827281314017899102130, 12.30631376923337532759570823072, 12.59059145512418500563991253843
