| L(s) = 1 | + 27·3-s − 240·5-s − 702·11-s + 7.91e3·13-s − 6.48e3·15-s − 3.40e3·17-s − 4.90e4·19-s + 1.15e4·23-s + 7.81e4·25-s − 1.96e4·27-s + 9.93e4·29-s − 1.13e5·31-s − 1.89e4·33-s + 6.68e4·37-s + 2.13e5·39-s + 7.21e5·41-s − 1.53e6·43-s − 1.34e6·47-s − 9.20e4·51-s − 3.58e5·53-s + 1.68e5·55-s − 1.32e6·57-s + 9.30e5·59-s − 1.31e6·61-s − 1.89e6·65-s − 1.89e6·67-s + 3.10e5·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.858·5-s − 0.159·11-s + 0.999·13-s − 0.495·15-s − 0.168·17-s − 1.64·19-s + 0.197·23-s + 25-s − 0.192·27-s + 0.756·29-s − 0.683·31-s − 0.0918·33-s + 0.217·37-s + 0.576·39-s + 1.63·41-s − 2.93·43-s − 1.88·47-s − 0.0971·51-s − 0.331·53-s + 0.136·55-s − 0.946·57-s + 0.589·59-s − 0.743·61-s − 0.858·65-s − 0.769·67-s + 0.113·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.006490385541\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.006490385541\) |
| \(L(\frac{9}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - p^{3} T + p^{6} T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 48 p T - 821 p^{2} T^{2} + 48 p^{8} T^{3} + p^{14} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 702 T - 18994367 T^{2} + 702 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 3958 T + p^{7} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 3408 T - 398724209 T^{2} + 3408 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 49036 T + 1510657557 T^{2} + 49036 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 11514 T - 3272253251 T^{2} - 11514 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 49662 T + p^{7} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 113320 T - 14671191711 T^{2} + 113320 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 66886 T - 90458140137 T^{2} - 66886 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 360900 T + p^{7} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 765292 T + p^{7} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 1344876 T + 1302068334913 T^{2} + 1344876 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 358962 T - 1045857422393 T^{2} + 358962 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 930528 T - 1622769126035 T^{2} - 930528 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 1318834 T - 1403419716465 T^{2} + 1318834 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 1893464 T - 2475505686027 T^{2} + 1893464 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 227994 T + p^{7} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 784934 T - 10431277134741 T^{2} - 784934 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 2100892 T - 14790161790495 T^{2} - 2100892 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 8629308 T + p^{7} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 5903100 T - 9384745285529 T^{2} - 5903100 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 773846 T + p^{7} T^{2} )^{2} \) |
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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
−10.28565362324233627190455855973, −9.353823132220563114240320320678, −8.631132802688597421831654762873, −8.368760544321691989042565484949, −8.345724912714822295177923069242, −7.69380658638387869247095417606, −7.14363531953807893442976682750, −6.60056421081598846498107604421, −6.42402888367121847406509468878, −5.74609149205146222878056715020, −5.15954334004229563405030312439, −4.47470294099587718420934114880, −4.32460038403474607941705563049, −3.63061136955871261108282482774, −3.18619571830922931356354109628, −2.80271193523160402905173852618, −2.02827724711382770840567649591, −1.55018273444343400705910321363, −0.940644247256849907555213777761, −0.01572704086729964171158144476,
0.01572704086729964171158144476, 0.940644247256849907555213777761, 1.55018273444343400705910321363, 2.02827724711382770840567649591, 2.80271193523160402905173852618, 3.18619571830922931356354109628, 3.63061136955871261108282482774, 4.32460038403474607941705563049, 4.47470294099587718420934114880, 5.15954334004229563405030312439, 5.74609149205146222878056715020, 6.42402888367121847406509468878, 6.60056421081598846498107604421, 7.14363531953807893442976682750, 7.69380658638387869247095417606, 8.345724912714822295177923069242, 8.368760544321691989042565484949, 8.631132802688597421831654762873, 9.353823132220563114240320320678, 10.28565362324233627190455855973
