| L(s) = 1 | + 128·2-s + 2.18e3·3-s + 1.63e4·4-s + 2.79e5·6-s + 5.11e5·7-s + 2.09e6·8-s + 4.78e6·9-s − 1.99e7·11-s + 3.58e7·12-s − 1.80e8·13-s + 6.55e7·14-s + 2.68e8·16-s + 5.65e8·17-s + 6.12e8·18-s − 2.16e9·19-s + 1.11e9·21-s − 2.55e9·22-s − 3.44e9·23-s + 4.58e9·24-s − 2.30e10·26-s + 1.04e10·27-s + 8.38e9·28-s + 5.88e10·29-s − 1.31e11·31-s + 3.43e10·32-s − 4.36e10·33-s + 7.24e10·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.234·7-s + 0.353·8-s + 1/3·9-s − 0.308·11-s + 0.288·12-s − 0.796·13-s + 0.166·14-s + 1/4·16-s + 0.334·17-s + 0.235·18-s − 0.556·19-s + 0.135·21-s − 0.218·22-s − 0.210·23-s + 0.204·24-s − 0.562·26-s + 0.192·27-s + 0.117·28-s + 0.633·29-s − 0.856·31-s + 0.176·32-s − 0.178·33-s + 0.236·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{7} T \) |
| 3 | \( 1 - p^{7} T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 73142 p T + p^{15} T^{2} \) |
| 11 | \( 1 + 1813418 p T + p^{15} T^{2} \) |
| 13 | \( 1 + 180100962 T + p^{15} T^{2} \) |
| 17 | \( 1 - 565654514 T + p^{15} T^{2} \) |
| 19 | \( 1 + 2169230520 T + p^{15} T^{2} \) |
| 23 | \( 1 + 3443072572 T + p^{15} T^{2} \) |
| 29 | \( 1 - 58843361520 T + p^{15} T^{2} \) |
| 31 | \( 1 + 131248934648 T + p^{15} T^{2} \) |
| 37 | \( 1 + 1018926148246 T + p^{15} T^{2} \) |
| 41 | \( 1 + 678311212798 T + p^{15} T^{2} \) |
| 43 | \( 1 - 1869778918508 T + p^{15} T^{2} \) |
| 47 | \( 1 + 2655546152576 T + p^{15} T^{2} \) |
| 53 | \( 1 - 7733409097998 T + p^{15} T^{2} \) |
| 59 | \( 1 - 12384358030090 T + p^{15} T^{2} \) |
| 61 | \( 1 + 28742040118198 T + p^{15} T^{2} \) |
| 67 | \( 1 + 62114558153336 T + p^{15} T^{2} \) |
| 71 | \( 1 - 17045715067452 T + p^{15} T^{2} \) |
| 73 | \( 1 - 94423385896028 T + p^{15} T^{2} \) |
| 79 | \( 1 - 5162001412320 T + p^{15} T^{2} \) |
| 83 | \( 1 + 388824931818532 T + p^{15} T^{2} \) |
| 89 | \( 1 - 64970078898710 T + p^{15} T^{2} \) |
| 97 | \( 1 + 424522131387176 T + p^{15} T^{2} \) |
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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
−9.908756422689079364879960174388, −8.686978501798226970572510652682, −7.68387736998302542568162645197, −6.77360299447531995249641327149, −5.47985867709444749680006705601, −4.55514892194849212236119816119, −3.47158730304053820464309971119, −2.47747290895136604635881319315, −1.53982859437516478884648952900, 0,
1.53982859437516478884648952900, 2.47747290895136604635881319315, 3.47158730304053820464309971119, 4.55514892194849212236119816119, 5.47985867709444749680006705601, 6.77360299447531995249641327149, 7.68387736998302542568162645197, 8.686978501798226970572510652682, 9.908756422689079364879960174388
