| L(s) = 1 | + 128·2-s − 1.30e3·3-s + 1.63e4·4-s − 7.81e4·5-s − 1.66e5·6-s − 9.07e4·7-s + 2.09e6·8-s − 1.26e7·9-s − 1.00e7·10-s − 6.94e7·11-s − 2.13e7·12-s − 3.70e7·13-s − 1.16e7·14-s + 1.01e8·15-s + 2.68e8·16-s − 1.37e9·17-s − 1.61e9·18-s − 5.06e9·19-s − 1.28e9·20-s + 1.18e8·21-s − 8.88e9·22-s − 1.23e10·23-s − 2.73e9·24-s + 6.10e9·25-s − 4.74e9·26-s + 3.51e10·27-s − 1.48e9·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.343·3-s + 1/2·4-s − 0.447·5-s − 0.243·6-s − 0.0416·7-s + 0.353·8-s − 0.881·9-s − 0.316·10-s − 1.07·11-s − 0.171·12-s − 0.163·13-s − 0.0294·14-s + 0.153·15-s + 1/4·16-s − 0.810·17-s − 0.623·18-s − 1.30·19-s − 0.223·20-s + 0.0143·21-s − 0.759·22-s − 0.755·23-s − 0.121·24-s + 1/5·25-s − 0.115·26-s + 0.646·27-s − 0.0208·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p^{7} T \) |
| 5 | \( 1 + p^{7} T \) |
| good | 3 | \( 1 + 434 p T + p^{15} T^{2} \) |
| 7 | \( 1 + 12958 p T + p^{15} T^{2} \) |
| 11 | \( 1 + 69411648 T + p^{15} T^{2} \) |
| 13 | \( 1 + 2850574 p T + p^{15} T^{2} \) |
| 17 | \( 1 + 1372027686 T + p^{15} T^{2} \) |
| 19 | \( 1 + 5068760620 T + p^{15} T^{2} \) |
| 23 | \( 1 + 12342632022 T + p^{15} T^{2} \) |
| 29 | \( 1 - 57825721470 T + p^{15} T^{2} \) |
| 31 | \( 1 - 233727970052 T + p^{15} T^{2} \) |
| 37 | \( 1 - 742247612954 T + p^{15} T^{2} \) |
| 41 | \( 1 + 772699832298 T + p^{15} T^{2} \) |
| 43 | \( 1 + 405994366942 T + p^{15} T^{2} \) |
| 47 | \( 1 - 1623010601574 T + p^{15} T^{2} \) |
| 53 | \( 1 + 101237724534 p T + p^{15} T^{2} \) |
| 59 | \( 1 - 9240158287140 T + p^{15} T^{2} \) |
| 61 | \( 1 - 944308151402 T + p^{15} T^{2} \) |
| 67 | \( 1 + 70567580292586 T + p^{15} T^{2} \) |
| 71 | \( 1 + 82534723020948 T + p^{15} T^{2} \) |
| 73 | \( 1 + 178432352158222 T + p^{15} T^{2} \) |
| 79 | \( 1 - 307261263603320 T + p^{15} T^{2} \) |
| 83 | \( 1 - 43159732395618 T + p^{15} T^{2} \) |
| 89 | \( 1 - 681677801811210 T + p^{15} T^{2} \) |
| 97 | \( 1 + 236520239800126 T + p^{15} T^{2} \) |
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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
−16.14991519545306930812106955401, −14.91053892401566569301805601460, −13.34471558792581281007342732326, −11.91899751599081035069024335113, −10.62767820029133970550889677192, −8.197770916599705618418060174180, −6.25699420939725110721559292712, −4.61879468917711307388318801525, −2.64409275326768320480884215330, 0,
2.64409275326768320480884215330, 4.61879468917711307388318801525, 6.25699420939725110721559292712, 8.197770916599705618418060174180, 10.62767820029133970550889677192, 11.91899751599081035069024335113, 13.34471558792581281007342732326, 14.91053892401566569301805601460, 16.14991519545306930812106955401
