| L(s) = 1 | + 1.02e3·2-s + 5.93e4·3-s + 1.04e6·4-s + 4.97e6·5-s + 6.07e7·6-s + 1.42e9·7-s + 1.07e9·8-s − 6.94e9·9-s + 5.09e9·10-s − 1.06e11·11-s + 6.21e10·12-s − 1.50e11·13-s + 1.46e12·14-s + 2.95e11·15-s + 1.09e12·16-s − 1.12e13·17-s − 7.10e12·18-s + 1.10e13·19-s + 5.21e12·20-s + 8.46e13·21-s − 1.09e14·22-s + 1.29e14·23-s + 6.36e13·24-s − 4.52e14·25-s − 1.53e14·26-s − 1.03e15·27-s + 1.49e15·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.579·3-s + 1/2·4-s + 0.227·5-s + 0.410·6-s + 1.90·7-s + 0.353·8-s − 0.663·9-s + 0.161·10-s − 1.24·11-s + 0.289·12-s − 0.302·13-s + 1.35·14-s + 0.132·15-s + 1/4·16-s − 1.34·17-s − 0.469·18-s + 0.412·19-s + 0.113·20-s + 1.10·21-s − 0.877·22-s + 0.651·23-s + 0.205·24-s − 0.948·25-s − 0.213·26-s − 0.964·27-s + 0.954·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(\approx\) |
\(2.875329661\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.875329661\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{10} T \) |
| 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
−23.62721432844297798285765870585, −21.43852141351332617357633491395, −20.27201337467160843800393087736, −17.74187424053087893891495975635, −15.08645291602186251139206673474, −13.72081476671816014358882137052, −11.24389269369499625431009592280, −8.088728315753170050801845056385, −5.00231375653705135805336012229, −2.28757101322957217312376409581,
2.28757101322957217312376409581, 5.00231375653705135805336012229, 8.088728315753170050801845056385, 11.24389269369499625431009592280, 13.72081476671816014358882137052, 15.08645291602186251139206673474, 17.74187424053087893891495975635, 20.27201337467160843800393087736, 21.43852141351332617357633491395, 23.62721432844297798285765870585
