| 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
| \(L(11)\) |
\(\approx\) |
\(0.002203273028\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.002203273028\) |
| \(L(\frac{23}{2})\) |
|
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
−10.77694963807242274022578208985, −9.706598614129879446766947430077, −8.836194694659863019614650555150, −7.70478566045072326031119294488, −6.51130292665536858947668614425, −5.52272980127657325310349169144, −3.84253651698750835727846059950, −3.04036908680483208183698655435, −2.16216656991606368104408793959, −0.01771173309823747651799937888,
0.01771173309823747651799937888, 2.16216656991606368104408793959, 3.04036908680483208183698655435, 3.84253651698750835727846059950, 5.52272980127657325310349169144, 6.51130292665536858947668614425, 7.70478566045072326031119294488, 8.836194694659863019614650555150, 9.706598614129879446766947430077, 10.77694963807242274022578208985
