| L(s) = 1 | − 128·2-s + 3.02e3·3-s + 1.63e4·4-s − 3.86e5·6-s − 2.09e6·8-s + 4.34e6·9-s + 3.87e7·11-s + 4.95e7·12-s + 2.68e8·16-s − 3.28e8·17-s − 5.56e8·18-s + 1.77e9·19-s − 4.95e9·22-s − 6.33e9·24-s + 6.10e9·25-s − 1.30e9·27-s − 3.43e10·32-s + 1.16e11·33-s + 4.20e10·34-s + 7.12e10·36-s − 2.27e11·38-s − 3.33e11·41-s − 4.95e11·43-s + 6.34e11·44-s + 8.11e11·48-s + 6.78e11·49-s − 7.81e11·50-s + ⋯ |
| L(s) = 1 | − 2-s + 1.38·3-s + 4-s − 1.38·6-s − 8-s + 0.909·9-s + 1.98·11-s + 1.38·12-s + 16-s − 0.800·17-s − 0.909·18-s + 1.99·19-s − 1.98·22-s − 1.38·24-s + 25-s − 0.125·27-s − 32-s + 2.74·33-s + 0.800·34-s + 0.909·36-s − 1.99·38-s − 1.71·41-s − 1.82·43-s + 1.98·44-s + 1.38·48-s + 49-s − 50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{15}{2})\) |
\(\approx\) |
\(1.900961425\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.900961425\) |
| \(L(8)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{7} T \) |
| good | 3 | \( 1 - 3022 T + p^{14} T^{2} \) |
| 5 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 7 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 11 | \( 1 - 38712254 T + p^{14} T^{2} \) |
| 13 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 17 | \( 1 + 328636222 T + p^{14} T^{2} \) |
| 19 | \( 1 - 1778973806 T + p^{14} T^{2} \) |
| 23 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 29 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 31 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 37 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 41 | \( 1 + 333393570766 T + p^{14} T^{2} \) |
| 43 | \( 1 + 495012562114 T + p^{14} T^{2} \) |
| 47 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 53 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 59 | \( 1 + 3914494552162 T + p^{14} T^{2} \) |
| 61 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 67 | \( 1 - 2711103884558 T + p^{14} T^{2} \) |
| 71 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 73 | \( 1 - 1579402558802 T + p^{14} T^{2} \) |
| 79 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 83 | \( 1 + 31146255762898 T + p^{14} T^{2} \) |
| 89 | \( 1 + 38433671549134 T + p^{14} T^{2} \) |
| 97 | \( 1 + 62815034524126 T + p^{14} T^{2} \) |
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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
−18.47905191911126664081832791761, −16.85997900015321141236509469629, −15.22712927524386064362087199261, −14.00024357206983060554327865514, −11.72142874505739255616761837470, −9.568716191943336008868342047222, −8.631829122108633054440403169790, −6.96000348790171253541822008014, −3.28655077430416968622719342561, −1.46323286457274976347416750291,
1.46323286457274976347416750291, 3.28655077430416968622719342561, 6.96000348790171253541822008014, 8.631829122108633054440403169790, 9.568716191943336008868342047222, 11.72142874505739255616761837470, 14.00024357206983060554327865514, 15.22712927524386064362087199261, 16.85997900015321141236509469629, 18.47905191911126664081832791761
