| L(s) = 1 | − 288·2-s + 1.28e5·3-s − 2.01e6·4-s − 2.16e7·5-s − 3.71e7·6-s + 1.18e9·8-s + 6.14e9·9-s + 6.23e9·10-s − 9.47e10·11-s − 2.59e11·12-s + 8.06e10·13-s − 2.78e12·15-s + 3.88e12·16-s − 3.05e12·17-s − 1.76e12·18-s + 7.92e12·19-s + 4.35e13·20-s + 2.72e13·22-s − 7.38e13·23-s + 1.52e14·24-s − 8.50e12·25-s − 2.32e13·26-s − 5.56e14·27-s − 4.25e15·29-s + 8.03e14·30-s − 1.90e15·31-s − 3.60e15·32-s + ⋯ |
| L(s) = 1 | − 0.198·2-s + 1.25·3-s − 0.960·4-s − 0.991·5-s − 0.250·6-s + 0.389·8-s + 0.587·9-s + 0.197·10-s − 1.10·11-s − 1.20·12-s + 0.162·13-s − 1.24·15-s + 0.882·16-s − 0.367·17-s − 0.116·18-s + 0.296·19-s + 0.951·20-s + 0.218·22-s − 0.371·23-s + 0.491·24-s − 0.0178·25-s − 0.0322·26-s − 0.520·27-s − 1.87·29-s + 0.248·30-s − 0.416·31-s − 0.565·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(\approx\) |
\(1.005861844\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.005861844\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 + 9 p^{5} T + p^{21} T^{2} \) |
| 3 | \( 1 - 4772 p^{3} T + p^{21} T^{2} \) |
| 5 | \( 1 + 865638 p^{2} T + p^{21} T^{2} \) |
| 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
−11.37892421208813126011940346077, −9.986656989511956973220452843249, −8.958616782966868885626062985606, −8.094218931444054281133026508008, −7.50902899829881004810291007066, −5.38866157041418963427654463106, −4.07838788535093264894691765572, −3.35948095517930697772030675914, −2.05584237510815072412726816380, −0.42599942385817125931319169735,
0.42599942385817125931319169735, 2.05584237510815072412726816380, 3.35948095517930697772030675914, 4.07838788535093264894691765572, 5.38866157041418963427654463106, 7.50902899829881004810291007066, 8.094218931444054281133026508008, 8.958616782966868885626062985606, 9.986656989511956973220452843249, 11.37892421208813126011940346077
