| L(s) = 1 | + 288·2-s − 2.01e6·4-s − 2.16e7·5-s − 7.68e8·7-s − 1.18e9·8-s − 6.23e9·10-s + 9.47e10·11-s − 8.06e10·13-s − 2.21e11·14-s + 3.88e12·16-s − 3.05e12·17-s − 7.92e12·19-s + 4.35e13·20-s + 2.72e13·22-s + 7.38e13·23-s − 8.50e12·25-s − 2.32e13·26-s + 1.54e15·28-s + 4.25e15·29-s + 1.90e15·31-s + 3.60e15·32-s − 8.79e14·34-s + 1.66e16·35-s + 2.21e16·37-s − 2.28e15·38-s + 2.56e16·40-s + 2.06e16·41-s + ⋯ |
| L(s) = 1 | + 0.198·2-s − 0.960·4-s − 0.991·5-s − 1.02·7-s − 0.389·8-s − 0.197·10-s + 1.10·11-s − 0.162·13-s − 0.204·14-s + 0.882·16-s − 0.367·17-s − 0.296·19-s + 0.951·20-s + 0.218·22-s + 0.371·23-s − 0.0178·25-s − 0.0322·26-s + 0.987·28-s + 1.87·29-s + 0.416·31-s + 0.565·32-s − 0.0730·34-s + 1.01·35-s + 0.758·37-s − 0.0589·38-s + 0.386·40-s + 0.239·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(\approx\) |
\(0.9491377251\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9491377251\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 - 9 p^{5} T + p^{21} T^{2} \) |
| 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
−15.94441105409599838473500695725, −14.52439059781124996688786149440, −13.04712111020834011132085532738, −11.82192346402638467055127219253, −9.771811744328875479261590906160, −8.430503189805388154554195761390, −6.53614326414672704890634055083, −4.48583664739598694492352633362, −3.34721222970318869243550169608, −0.61755191388686341345947756013,
0.61755191388686341345947756013, 3.34721222970318869243550169608, 4.48583664739598694492352633362, 6.53614326414672704890634055083, 8.430503189805388154554195761390, 9.771811744328875479261590906160, 11.82192346402638467055127219253, 13.04712111020834011132085532738, 14.52439059781124996688786149440, 15.94441105409599838473500695725
