| L(s) = 1 | − 3-s + 5-s + 7-s + 9-s + 11-s + 13-s − 15-s + 17-s + 19-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 + 49-s − 51-s − 53-s + 55-s − 57-s + ⋯ |
| L(s) = 1 | − 3-s + 5-s + 7-s + 9-s + 11-s + 13-s − 15-s + 17-s + 19-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 + 49-s − 51-s − 53-s + 55-s − 57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 376 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 376 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.459062873\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.459062873\) |
| \(L(1)\) |
\(\approx\) |
\(1.296122164\) |
| \(L(1)\) |
\(\approx\) |
\(1.296122164\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 47 | \( 1 \) |
| good | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| 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
−24.34558751245487597712159219539, −23.51860699805304730722646395675, −22.52870013639146088332874114546, −21.78272024149242251890653426406, −21.0777720707793354067562806575, −20.23628900720677836640073092565, −18.69082520010943670410857793262, −18.06129140433865603381008124942, −17.37017740543009275860559607016, −16.62064495187312740103043131781, −15.66723875406575564585141686200, −14.27875279205308933604997302389, −13.8261511013688434841316920695, −12.46378967961980692788802999348, −11.715552954314817866357401044599, −10.80714216722378978975589845712, −9.952890420986384996367644213729, −8.93014794515243194058166106566, −7.65061699566650475621003494032, −6.46649995240677302465019574625, −5.70525083839611455638672145240, −4.87044439908837907816198187545, −3.61134668798444228469816939521, −1.719460147707985193868120111502, −1.09416802072735167035160078763,
1.09416802072735167035160078763, 1.719460147707985193868120111502, 3.61134668798444228469816939521, 4.87044439908837907816198187545, 5.70525083839611455638672145240, 6.46649995240677302465019574625, 7.65061699566650475621003494032, 8.93014794515243194058166106566, 9.952890420986384996367644213729, 10.80714216722378978975589845712, 11.715552954314817866357401044599, 12.46378967961980692788802999348, 13.8261511013688434841316920695, 14.27875279205308933604997302389, 15.66723875406575564585141686200, 16.62064495187312740103043131781, 17.37017740543009275860559607016, 18.06129140433865603381008124942, 18.69082520010943670410857793262, 20.23628900720677836640073092565, 21.0777720707793354067562806575, 21.78272024149242251890653426406, 22.52870013639146088332874114546, 23.51860699805304730722646395675, 24.34558751245487597712159219539
