| L(s) = 1 | − 5.93e4·3-s − 4.97e6·5-s + 1.42e9·7-s − 6.94e9·9-s + 1.06e11·11-s + 1.50e11·13-s + 2.95e11·15-s − 1.12e13·17-s − 1.10e13·19-s − 8.46e13·21-s + 1.29e14·23-s − 4.52e14·25-s + 1.03e15·27-s − 2.38e15·29-s − 8.78e14·31-s − 6.33e15·33-s − 7.10e15·35-s − 3.11e16·37-s − 8.90e15·39-s − 2.46e16·41-s + 1.33e17·43-s + 3.45e16·45-s − 1.92e17·47-s + 1.47e18·49-s + 6.64e17·51-s + 5.94e17·53-s − 5.31e17·55-s + ⋯ |
| L(s) = 1 | − 0.579·3-s − 0.227·5-s + 1.90·7-s − 0.663·9-s + 1.24·11-s + 0.302·13-s + 0.132·15-s − 1.34·17-s − 0.412·19-s − 1.10·21-s + 0.651·23-s − 0.948·25-s + 0.964·27-s − 1.05·29-s − 0.192·31-s − 0.719·33-s − 0.435·35-s − 1.06·37-s − 0.175·39-s − 0.286·41-s + 0.941·43-s + 0.151·45-s − 0.533·47-s + 2.64·49-s + 0.781·51-s + 0.466·53-s − 0.282·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 19772 p T + p^{21} T^{2} \) |
| 5 | \( 1 + 199014 p^{2} T + p^{21} T^{2} \) |
| 7 | \( 1 - 203917976 p T + p^{21} T^{2} \) |
| 11 | \( 1 - 9706172268 p T + p^{21} T^{2} \) |
| 13 | \( 1 - 11550043498 p T + p^{21} T^{2} \) |
| 17 | \( 1 + 659057690574 p T + p^{21} T^{2} \) |
| 19 | \( 1 + 580213471340 p T + p^{21} T^{2} \) |
| 23 | \( 1 - 129502845739896 T + p^{21} T^{2} \) |
| 29 | \( 1 + 2382370826608110 T + p^{21} T^{2} \) |
| 31 | \( 1 + 878552957377888 T + p^{21} T^{2} \) |
| 37 | \( 1 + 31130005856560022 T + p^{21} T^{2} \) |
| 41 | \( 1 + 24612925945718838 T + p^{21} T^{2} \) |
| 43 | \( 1 - 133386119963316484 T + p^{21} T^{2} \) |
| 47 | \( 1 + 192524017446421008 T + p^{21} T^{2} \) |
| 53 | \( 1 - 594166360130841114 T + p^{21} T^{2} \) |
| 59 | \( 1 - 2955954134483673780 T + p^{21} T^{2} \) |
| 61 | \( 1 + 7984150090052846222 T + p^{21} T^{2} \) |
| 67 | \( 1 + 4837041486709240052 T + p^{21} T^{2} \) |
| 71 | \( 1 - 8849017338933008232 T + p^{21} T^{2} \) |
| 73 | \( 1 - 36684416180434869866 T + p^{21} T^{2} \) |
| 79 | \( 1 - 33840609578636773520 T + p^{21} T^{2} \) |
| 83 | \( 1 + \)\(20\!\cdots\!16\)\( T + p^{21} T^{2} \) |
| 89 | \( 1 + 41024056743692272710 T + p^{21} T^{2} \) |
| 97 | \( 1 + \)\(72\!\cdots\!98\)\( T + p^{21} T^{2} \) |
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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
−10.94127019584115575902390159166, −9.040993567860133677288882039921, −8.311481273549829285239253179621, −7.04322013566529086083319224450, −5.83442952754921982199426128030, −4.80044874855954074174382848733, −3.89839871323133709723652546423, −2.13292530004400053937836480849, −1.25644636514673892338885272659, 0,
1.25644636514673892338885272659, 2.13292530004400053937836480849, 3.89839871323133709723652546423, 4.80044874855954074174382848733, 5.83442952754921982199426128030, 7.04322013566529086083319224450, 8.311481273549829285239253179621, 9.040993567860133677288882039921, 10.94127019584115575902390159166
