| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(\approx\) |
\(0.7976700413\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7976700413\) |
| \(L(\frac{23}{2})\) |
|
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 | \( 1 + \)\(72\!\cdots\!98\)\( T + p^{21} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.00681583481980551675435068580, −12.73240325302763704159033384983, −11.57344825690848785073278693714, −10.01156383304620894943020761739, −8.926541176739463283638910951499, −6.66393719325511045204620242673, −6.02612619159973860274284627817, −4.04184324324703540106196216822, −2.55300370275496585000533255255, −0.50321077040495664645892968891,
0.50321077040495664645892968891, 2.55300370275496585000533255255, 4.04184324324703540106196216822, 6.02612619159973860274284627817, 6.66393719325511045204620242673, 8.926541176739463283638910951499, 10.01156383304620894943020761739, 11.57344825690848785073278693714, 12.73240325302763704159033384983, 14.00681583481980551675435068580
