| L(s) = 1 | − 0.843·2-s − 511.·4-s − 8.71e3·7-s + 863.·8-s − 4.45e4·11-s + 2.14e4·13-s + 7.35e3·14-s + 2.61e5·16-s + 3.00e5·17-s + 5.65e5·19-s + 3.76e4·22-s + 9.50e5·23-s − 1.80e4·26-s + 4.45e6·28-s + 8.03e5·29-s − 1.99e6·31-s − 6.62e5·32-s − 2.53e5·34-s − 9.53e6·37-s − 4.77e5·38-s + 2.54e7·41-s + 2.32e7·43-s + 2.27e7·44-s − 8.02e5·46-s − 3.77e7·47-s + 3.55e7·49-s − 1.09e7·52-s + ⋯ |
| L(s) = 1 | − 0.0372·2-s − 0.998·4-s − 1.37·7-s + 0.0745·8-s − 0.917·11-s + 0.208·13-s + 0.0511·14-s + 0.995·16-s + 0.871·17-s + 0.995·19-s + 0.0342·22-s + 0.708·23-s − 0.00776·26-s + 1.36·28-s + 0.210·29-s − 0.388·31-s − 0.111·32-s − 0.0325·34-s − 0.836·37-s − 0.0371·38-s + 1.40·41-s + 1.03·43-s + 0.916·44-s − 0.0264·46-s − 1.12·47-s + 0.881·49-s − 0.207·52-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
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 + 0.843T + 512T^{2} \) |
| 7 | \( 1 + 8.71e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 4.45e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 2.14e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 3.00e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 5.65e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 9.50e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 8.03e5T + 1.45e13T^{2} \) |
| 31 | \( 1 + 1.99e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 9.53e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.54e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.32e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 3.77e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 4.79e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 7.00e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.26e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.66e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 6.59e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.47e7T + 5.88e16T^{2} \) |
| 79 | \( 1 + 4.66e7T + 1.19e17T^{2} \) |
| 83 | \( 1 - 2.01e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 5.54e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 3.39e8T + 7.60e17T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.853066339729584810273237080267, −9.392725224713822378312224583736, −8.226538077149830169588942897307, −7.22094181475739220420563863061, −5.89423011455646446628517085111, −5.05326049288656395227540833893, −3.67544521448636128505369144159, −2.89059338514766843404693408644, −0.990878349365282007152932857581, 0,
0.990878349365282007152932857581, 2.89059338514766843404693408644, 3.67544521448636128505369144159, 5.05326049288656395227540833893, 5.89423011455646446628517085111, 7.22094181475739220420563863061, 8.226538077149830169588942897307, 9.392725224713822378312224583736, 9.853066339729584810273237080267
