L(s) = 1 | + 3-s + 5-s − 7-s + 9-s − 11-s − 13-s + 15-s + 17-s − 21-s − 23-s + 25-s + 27-s − 29-s + 31-s − 33-s − 35-s − 37-s − 39-s − 41-s − 43-s + 45-s − 47-s + 49-s + 51-s − 53-s − 55-s + 59-s + ⋯ |
L(s) = 1 | + 3-s + 5-s − 7-s + 9-s − 11-s − 13-s + 15-s + 17-s − 21-s − 23-s + 25-s + 27-s − 29-s + 31-s − 33-s − 35-s − 37-s − 39-s − 41-s − 43-s + 45-s − 47-s + 49-s + 51-s − 53-s − 55-s + 59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.287000630\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.287000630\) |
\(L(1)\) |
\(\approx\) |
\(1.337249851\) |
\(L(1)\) |
\(\approx\) |
\(1.337249851\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 - T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−31.742411544925891920425159430940, −30.09238558617870596295572083519, −29.41486871134307180863819261945, −28.25400108729443433901288765115, −26.61491924449366982256966535766, −25.92231168250869541964468098425, −25.12479853680107620905462960975, −24.000727185523306388055498829581, −22.413981882951850770734071857, −21.38540608952082941863952658631, −20.436942974250486087195291827645, −19.23670270979561450701239235769, −18.307920228896712125168295192920, −16.82600648030593608040981992716, −15.606521894277928419526228415964, −14.35097234624181254879080448543, −13.364122447975648155484403913337, −12.468799648816489039569645571430, −10.08539409531236167027430699124, −9.727828837994235452232957984486, −8.166301389804727017420484256839, −6.80525558008000178501586865609, −5.25059192447677962991272377620, −3.30247173383803505013623446369, −2.149932773489239013639461398865,
2.149932773489239013639461398865, 3.30247173383803505013623446369, 5.25059192447677962991272377620, 6.80525558008000178501586865609, 8.166301389804727017420484256839, 9.727828837994235452232957984486, 10.08539409531236167027430699124, 12.468799648816489039569645571430, 13.364122447975648155484403913337, 14.35097234624181254879080448543, 15.606521894277928419526228415964, 16.82600648030593608040981992716, 18.307920228896712125168295192920, 19.23670270979561450701239235769, 20.436942974250486087195291827645, 21.38540608952082941863952658631, 22.413981882951850770734071857, 24.000727185523306388055498829581, 25.12479853680107620905462960975, 25.92231168250869541964468098425, 26.61491924449366982256966535766, 28.25400108729443433901288765115, 29.41486871134307180863819261945, 30.09238558617870596295572083519, 31.742411544925891920425159430940