| L(s) = 1 | − 76·3-s + 250·5-s − 796·7-s + 2.36e3·9-s + 560·11-s + 4.47e3·13-s − 1.90e4·15-s + 2.95e4·17-s − 1.95e4·19-s + 6.04e4·21-s − 1.29e5·23-s + 4.68e4·25-s − 8.62e4·27-s − 2.11e5·29-s − 1.65e5·31-s − 4.25e4·33-s − 1.99e5·35-s − 2.92e5·37-s − 3.40e5·39-s − 8.51e5·41-s + 1.97e5·43-s + 5.90e5·45-s + 3.41e4·47-s − 1.05e6·49-s − 2.24e6·51-s − 8.76e5·53-s + 1.40e5·55-s + ⋯ |
| L(s) = 1 | − 1.62·3-s + 0.894·5-s − 0.877·7-s + 1.08·9-s + 0.126·11-s + 0.565·13-s − 1.45·15-s + 1.45·17-s − 0.654·19-s + 1.42·21-s − 2.22·23-s + 3/5·25-s − 0.843·27-s − 1.60·29-s − 0.996·31-s − 0.206·33-s − 0.784·35-s − 0.948·37-s − 0.918·39-s − 1.92·41-s + 0.379·43-s + 0.965·45-s + 0.0479·47-s − 1.27·49-s − 2.36·51-s − 0.808·53-s + 0.113·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - p^{3} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 76 T + 1138 p T^{2} + 76 p^{7} T^{3} + p^{14} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 796 T + 1687694 T^{2} + 796 p^{7} T^{3} + p^{14} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 560 T + 5579446 T^{2} - 560 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4476 T + 88618382 T^{2} - 4476 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 29524 T + 1024708486 T^{2} - 29524 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 19560 T + 450646342 T^{2} + 19560 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 129796 T + 9416757742 T^{2} + 129796 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 211060 T + 35788164814 T^{2} + 211060 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 165352 T + 55844000702 T^{2} + 165352 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 292100 T + 100816677662 T^{2} + 292100 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 20764 p T + 533529345910 T^{2} + 20764 p^{8} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 197812 T + 500337854246 T^{2} - 197812 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 34164 T + 1008165035550 T^{2} - 34164 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 876404 T + 2203297594078 T^{2} + 876404 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 2088504 T + 1860966606646 T^{2} - 2088504 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 1409708 T + 4803809602382 T^{2} - 1409708 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 2882076 T + 14114698855414 T^{2} - 2882076 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 9819448 T + 42279153239662 T^{2} + 9819448 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 189404 T + 1441930580342 T^{2} + 189404 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 2028496 T + 25840354532958 T^{2} - 2028496 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 1393332 T + 43586027298166 T^{2} - 1393332 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 5688820 T + 96481985626742 T^{2} - 5688820 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 20045996 T + 226498473606630 T^{2} + 20045996 p^{7} T^{3} + p^{14} T^{4} \) |
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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
−12.60724952735029441018714488585, −12.09813910357064843944301589020, −11.59509736492515650204509036672, −11.13419917629649867373659916444, −10.29719058868633205078897531774, −10.13328968909981804277531004366, −9.577928780683080045190183440432, −8.855170092039988511204173470415, −8.018481496372073884409201092782, −7.27369750898504487498242918818, −6.33697736417663820911501333956, −6.23582737685781654097497858106, −5.47277580415748866847152983011, −5.26553616840009457166395855891, −3.92484378533459227069141389176, −3.42827266847628685852255906599, −2.02787948806047791419936133825, −1.39901915841043927628217219546, 0, 0,
1.39901915841043927628217219546, 2.02787948806047791419936133825, 3.42827266847628685852255906599, 3.92484378533459227069141389176, 5.26553616840009457166395855891, 5.47277580415748866847152983011, 6.23582737685781654097497858106, 6.33697736417663820911501333956, 7.27369750898504487498242918818, 8.018481496372073884409201092782, 8.855170092039988511204173470415, 9.577928780683080045190183440432, 10.13328968909981804277531004366, 10.29719058868633205078897531774, 11.13419917629649867373659916444, 11.59509736492515650204509036672, 12.09813910357064843944301589020, 12.60724952735029441018714488585
