| L(s) = 1 | − 999·2-s + 7.35e5·4-s − 4.03e7·7-s − 4.73e8·8-s + 3.87e8·9-s − 2.87e9·11-s + 4.03e10·14-s + 2.79e11·16-s − 3.87e11·18-s + 2.87e12·22-s − 1.65e12·23-s + 3.81e12·25-s − 2.96e13·28-s + 2.83e13·29-s − 1.55e14·32-s + 2.85e14·36-s + 2.57e14·37-s + 3.71e14·43-s − 2.11e15·44-s + 1.65e15·46-s + 1.62e15·49-s − 3.81e15·50-s − 3.22e15·53-s + 1.90e16·56-s − 2.82e16·58-s − 1.56e16·63-s + 8.20e16·64-s + ⋯ |
| L(s) = 1 | − 1.95·2-s + 2.80·4-s − 7-s − 3.52·8-s + 9-s − 1.21·11-s + 1.95·14-s + 4.07·16-s − 1.95·18-s + 2.37·22-s − 0.917·23-s + 25-s − 2.80·28-s + 1.95·29-s − 4.42·32-s + 2.80·36-s + 1.98·37-s + 0.738·43-s − 3.42·44-s + 1.79·46-s + 49-s − 1.95·50-s − 0.975·53-s + 3.52·56-s − 3.80·58-s − 63-s + 4.55·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{19}{2})\) |
\(\approx\) |
\(0.5902306649\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5902306649\) |
| \(L(10)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + p^{9} T \) |
| good | 2 | \( 1 + 999 T + p^{18} T^{2} \) |
| 3 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 5 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 11 | \( 1 + 2874798918 T + p^{18} T^{2} \) |
| 13 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 17 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 19 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 23 | \( 1 + 1653377641326 T + p^{18} T^{2} \) |
| 29 | \( 1 - 28312199721738 T + p^{18} T^{2} \) |
| 31 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 37 | \( 1 - 257341038312346 T + p^{18} T^{2} \) |
| 41 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 43 | \( 1 - 371356855294714 T + p^{18} T^{2} \) |
| 47 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 53 | \( 1 + 3220047777380166 T + p^{18} T^{2} \) |
| 59 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 61 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 67 | \( 1 - 14477358615829706 T + p^{18} T^{2} \) |
| 71 | \( 1 - 78927867169189362 T + p^{18} T^{2} \) |
| 73 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 79 | \( 1 - 124983776054130338 T + p^{18} T^{2} \) |
| 83 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 89 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 97 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
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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
−18.01505793266917478210896988195, −16.33487990209612522573481966410, −15.62178929387563688808090427976, −12.54571710415383089787335611487, −10.57279449621587018049705268186, −9.630675643764077406567311149828, −7.939578229223884769462860996852, −6.53561299337365954653444027232, −2.61550544577775513203566639916, −0.74533714115285948243811451094,
0.74533714115285948243811451094, 2.61550544577775513203566639916, 6.53561299337365954653444027232, 7.939578229223884769462860996852, 9.630675643764077406567311149828, 10.57279449621587018049705268186, 12.54571710415383089787335611487, 15.62178929387563688808090427976, 16.33487990209612522573481966410, 18.01505793266917478210896988195
