| L(s) = 1 | − 7.16e4·3-s − 2.86e7·5-s + 8.53e8·7-s − 5.33e9·9-s − 8.67e10·11-s − 8.95e11·13-s + 2.05e12·15-s + 3.25e12·17-s − 2.30e13·19-s − 6.10e13·21-s − 1.46e14·23-s + 3.46e14·25-s + 1.13e15·27-s − 7.34e14·29-s + 3.14e15·31-s + 6.21e15·33-s − 2.44e16·35-s − 1.29e16·37-s + 6.41e16·39-s + 4.57e16·41-s + 2.40e16·43-s + 1.53e17·45-s + 4.49e17·47-s + 1.69e17·49-s − 2.33e17·51-s + 2.06e18·53-s + 2.48e18·55-s + ⋯ |
| L(s) = 1 | − 0.700·3-s − 1.31·5-s + 1.14·7-s − 0.509·9-s − 1.00·11-s − 1.80·13-s + 0.919·15-s + 0.391·17-s − 0.861·19-s − 0.799·21-s − 0.737·23-s + 0.726·25-s + 1.05·27-s − 0.324·29-s + 0.689·31-s + 0.705·33-s − 1.50·35-s − 0.443·37-s + 1.26·39-s + 0.531·41-s + 0.169·43-s + 0.669·45-s + 1.24·47-s + 0.303·49-s − 0.274·51-s + 1.62·53-s + 1.32·55-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
| \(L(11)\) |
\(\approx\) |
\(0.5299471629\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5299471629\) |
| \(L(\frac{23}{2})\) |
|
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 | \( 1 + 884 p^{4} T + p^{21} T^{2} \) |
| 5 | \( 1 + 5738754 p T + p^{21} T^{2} \) |
| 7 | \( 1 - 121886056 p T + p^{21} T^{2} \) |
| 11 | \( 1 + 7884652692 p T + p^{21} T^{2} \) |
| 13 | \( 1 + 895323442786 T + p^{21} T^{2} \) |
| 17 | \( 1 - 191621576754 p T + p^{21} T^{2} \) |
| 19 | \( 1 + 1212235139180 p T + p^{21} T^{2} \) |
| 23 | \( 1 + 146495714575224 T + p^{21} T^{2} \) |
| 29 | \( 1 + 734051633521170 T + p^{21} T^{2} \) |
| 31 | \( 1 - 3146664162057568 T + p^{21} T^{2} \) |
| 37 | \( 1 + 12963813600992362 T + p^{21} T^{2} \) |
| 41 | \( 1 - 45714648841476042 T + p^{21} T^{2} \) |
| 43 | \( 1 - 24073607797047556 T + p^{21} T^{2} \) |
| 47 | \( 1 - 449991905173684752 T + p^{21} T^{2} \) |
| 53 | \( 1 - 2064837217091540454 T + p^{21} T^{2} \) |
| 59 | \( 1 - 3780497099978396340 T + p^{21} T^{2} \) |
| 61 | \( 1 + 7619813346829729138 T + p^{21} T^{2} \) |
| 67 | \( 1 - 18791158016925310732 T + p^{21} T^{2} \) |
| 71 | \( 1 - 4526486567453771928 T + p^{21} T^{2} \) |
| 73 | \( 1 + 25571455286910443926 T + p^{21} T^{2} \) |
| 79 | \( 1 + 99336442530925070480 T + p^{21} T^{2} \) |
| 83 | \( 1 + 2958180217887529284 T + p^{21} T^{2} \) |
| 89 | \( 1 - \)\(11\!\cdots\!90\)\( T + p^{21} T^{2} \) |
| 97 | \( 1 + \)\(56\!\cdots\!02\)\( T + p^{21} T^{2} \) |
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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
−14.41484334951564554656555102498, −12.33661038945480017351573824677, −11.59376098474558545627292353354, −10.46309627397745477004350978574, −8.285379533975298814160397354086, −7.40913632847644029271463399932, −5.39008946160630974729409194745, −4.37693565980633673693388743373, −2.46600309323073090837938644433, −0.41221677108527939841996478809,
0.41221677108527939841996478809, 2.46600309323073090837938644433, 4.37693565980633673693388743373, 5.39008946160630974729409194745, 7.40913632847644029271463399932, 8.285379533975298814160397354086, 10.46309627397745477004350978574, 11.59376098474558545627292353354, 12.33661038945480017351573824677, 14.41484334951564554656555102498
