| L(s) = 1 | − 36·2-s + 784·4-s + 4.48e3·7-s − 9.79e3·8-s − 1.47e3·11-s + 1.51e5·13-s − 1.61e5·14-s − 4.88e4·16-s + 1.08e5·17-s + 5.93e5·19-s + 5.31e4·22-s − 9.69e5·23-s − 5.45e6·26-s + 3.51e6·28-s + 6.64e6·29-s + 7.07e6·31-s + 6.77e6·32-s − 3.89e6·34-s + 7.47e6·37-s − 2.13e7·38-s + 4.35e6·41-s + 4.35e6·43-s − 1.15e6·44-s + 3.49e7·46-s + 2.83e7·47-s − 2.02e7·49-s + 1.18e8·52-s + ⋯ |
| L(s) = 1 | − 1.59·2-s + 1.53·4-s + 0.705·7-s − 0.845·8-s − 0.0303·11-s + 1.47·13-s − 1.12·14-s − 0.186·16-s + 0.314·17-s + 1.04·19-s + 0.0483·22-s − 0.722·23-s − 2.34·26-s + 1.07·28-s + 1.74·29-s + 1.37·31-s + 1.14·32-s − 0.499·34-s + 0.655·37-s − 1.66·38-s + 0.240·41-s + 0.194·43-s − 0.0465·44-s + 1.14·46-s + 0.846·47-s − 0.502·49-s + 2.25·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.380270220\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.380270220\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 9 p^{2} T + p^{9} T^{2} \) |
| 7 | \( 1 - 640 p T + p^{9} T^{2} \) |
| 11 | \( 1 + 1476 T + p^{9} T^{2} \) |
| 13 | \( 1 - 151522 T + p^{9} T^{2} \) |
| 17 | \( 1 - 108162 T + p^{9} T^{2} \) |
| 19 | \( 1 - 593084 T + p^{9} T^{2} \) |
| 23 | \( 1 + 969480 T + p^{9} T^{2} \) |
| 29 | \( 1 - 6642522 T + p^{9} T^{2} \) |
| 31 | \( 1 - 7070600 T + p^{9} T^{2} \) |
| 37 | \( 1 - 7472410 T + p^{9} T^{2} \) |
| 41 | \( 1 - 4350150 T + p^{9} T^{2} \) |
| 43 | \( 1 - 4358716 T + p^{9} T^{2} \) |
| 47 | \( 1 - 28309248 T + p^{9} T^{2} \) |
| 53 | \( 1 - 16111710 T + p^{9} T^{2} \) |
| 59 | \( 1 - 86075964 T + p^{9} T^{2} \) |
| 61 | \( 1 - 32213918 T + p^{9} T^{2} \) |
| 67 | \( 1 + 99531452 T + p^{9} T^{2} \) |
| 71 | \( 1 - 44170488 T + p^{9} T^{2} \) |
| 73 | \( 1 - 23560630 T + p^{9} T^{2} \) |
| 79 | \( 1 + 401754760 T + p^{9} T^{2} \) |
| 83 | \( 1 + 744528708 T + p^{9} T^{2} \) |
| 89 | \( 1 + 769871034 T + p^{9} T^{2} \) |
| 97 | \( 1 + 907130882 T + p^{9} T^{2} \) |
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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
−10.41260544035081236826563898597, −9.635252035028591570830205709268, −8.461042018751187007838130770960, −8.117765068104149104810662538821, −6.95742095036856292366808713230, −5.85268182220873976865279425736, −4.35025159794803078809689263062, −2.77880112413994984880096300320, −1.40854006742784389331510945235, −0.805855691193862578408976216230,
0.805855691193862578408976216230, 1.40854006742784389331510945235, 2.77880112413994984880096300320, 4.35025159794803078809689263062, 5.85268182220873976865279425736, 6.95742095036856292366808713230, 8.117765068104149104810662538821, 8.461042018751187007838130770960, 9.635252035028591570830205709268, 10.41260544035081236826563898597
