| L(s) = 1 | + 1.28e5·3-s + 2.16e7·5-s + 7.68e8·7-s + 6.14e9·9-s + 9.47e10·11-s − 8.06e10·13-s + 2.78e12·15-s + 3.05e12·17-s + 7.92e12·19-s + 9.89e13·21-s + 7.38e13·23-s − 8.50e12·25-s − 5.56e14·27-s − 4.25e15·29-s − 1.90e15·31-s + 1.22e16·33-s + 1.66e16·35-s + 2.21e16·37-s − 1.03e16·39-s − 2.06e16·41-s + 1.93e17·43-s + 1.32e17·45-s − 1.46e17·47-s + 3.13e16·49-s + 3.93e17·51-s + 2.03e18·53-s + 2.04e18·55-s + ⋯ |
| L(s) = 1 | + 1.25·3-s + 0.991·5-s + 1.02·7-s + 0.587·9-s + 1.10·11-s − 0.162·13-s + 1.24·15-s + 0.367·17-s + 0.296·19-s + 1.29·21-s + 0.371·23-s − 0.0178·25-s − 0.520·27-s − 1.87·29-s − 0.416·31-s + 1.38·33-s + 1.01·35-s + 0.758·37-s − 0.204·39-s − 0.239·41-s + 1.36·43-s + 0.581·45-s − 0.407·47-s + 0.0562·49-s + 0.462·51-s + 1.60·53-s + 1.09·55-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
| \(L(11)\) |
\(\approx\) |
\(4.628843537\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.628843537\) |
| \(L(\frac{23}{2})\) |
|
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 | \( 1 - 4772 p^{3} T + p^{21} T^{2} \) |
| 5 | \( 1 - 865638 p^{2} T + p^{21} T^{2} \) |
| 7 | \( 1 - 109725544 p T + p^{21} T^{2} \) |
| 11 | \( 1 - 94724929188 T + p^{21} T^{2} \) |
| 13 | \( 1 + 6201676138 p T + p^{21} T^{2} \) |
| 17 | \( 1 - 179546054706 p T + p^{21} T^{2} \) |
| 19 | \( 1 - 416883597460 p T + p^{21} T^{2} \) |
| 23 | \( 1 - 73845437470344 T + p^{21} T^{2} \) |
| 29 | \( 1 + 4253031736469010 T + p^{21} T^{2} \) |
| 31 | \( 1 + 1900541176310432 T + p^{21} T^{2} \) |
| 37 | \( 1 - 22191429912035222 T + p^{21} T^{2} \) |
| 41 | \( 1 + 20622803144546358 T + p^{21} T^{2} \) |
| 43 | \( 1 - 193605854685795844 T + p^{21} T^{2} \) |
| 47 | \( 1 + 146960504315611632 T + p^{21} T^{2} \) |
| 53 | \( 1 - 2038267110310687206 T + p^{21} T^{2} \) |
| 59 | \( 1 - 5975882742742352820 T + p^{21} T^{2} \) |
| 61 | \( 1 - 6190617154478149262 T + p^{21} T^{2} \) |
| 67 | \( 1 + 16961315295446680052 T + p^{21} T^{2} \) |
| 71 | \( 1 - 5632758963952293528 T + p^{21} T^{2} \) |
| 73 | \( 1 + 43284759511102937494 T + p^{21} T^{2} \) |
| 79 | \( 1 - 51264938664949064560 T + p^{21} T^{2} \) |
| 83 | \( 1 + 48911854702961049156 T + p^{21} T^{2} \) |
| 89 | \( 1 + \)\(50\!\cdots\!30\)\( T + p^{21} T^{2} \) |
| 97 | \( 1 - \)\(80\!\cdots\!82\)\( T + p^{21} T^{2} \) |
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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
−14.32368890264709780001280582216, −13.23670293184493619363397638593, −11.44236508297734892489302307278, −9.688799489044643196416150872307, −8.782511875212998542424449104675, −7.44044693381564311714203796866, −5.61052467767356011446947095405, −3.87451363565304382098854923957, −2.32446920990914596452098697305, −1.38570023789513565090355625811,
1.38570023789513565090355625811, 2.32446920990914596452098697305, 3.87451363565304382098854923957, 5.61052467767356011446947095405, 7.44044693381564311714203796866, 8.782511875212998542424449104675, 9.688799489044643196416150872307, 11.44236508297734892489302307278, 13.23670293184493619363397638593, 14.32368890264709780001280582216
