L(s) = 1 | + 2-s + 4-s + 5-s − 7-s + 8-s + 10-s + 11-s − 13-s − 14-s + 16-s − 17-s + 19-s + 20-s + 22-s − 23-s + 25-s − 26-s − 28-s − 29-s + 31-s + 32-s − 34-s − 35-s − 37-s + 38-s + 40-s − 41-s + ⋯ |
L(s) = 1 | + 2-s + 4-s + 5-s − 7-s + 8-s + 10-s + 11-s − 13-s − 14-s + 16-s − 17-s + 19-s + 20-s + 22-s − 23-s + 25-s − 26-s − 28-s − 29-s + 31-s + 32-s − 34-s − 35-s − 37-s + 38-s + 40-s − 41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5241 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5241 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(4.253536826\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.253536826\) |
\(L(1)\) |
\(\approx\) |
\(2.227664509\) |
\(L(1)\) |
\(\approx\) |
\(2.227664509\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 1747 | \( 1 \) |
good | 2 | \( 1 + T \) |
| 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 | \( 1 + T \) |
| 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
−17.70881198383319493853119635200, −17.198331100419834657516944168802, −16.59397104525626087049685641198, −15.87690844054797963448457424082, −15.27821527052289436983144672482, −14.41258776041154682294457508840, −13.95317736486632522114919534573, −13.38618182643212085511306068648, −12.75487746529069046323826835840, −12.04014671948622788740813324819, −11.57700729002383477774333791990, −10.49962611820311332142964372491, −9.950082838506073479573382514632, −9.39615972506210710887721056787, −8.58485465388722994339518214773, −7.29361255624994724676624978452, −6.904495287392644065696186137477, −6.20713541083947351938607876353, −5.62859583564405582061301903309, −4.90744494731475789598142654482, −4.01401067683220677870862320353, −3.3940051924703167013653131768, −2.422046696332539119013647260240, −2.04025144247440599431011115088, −0.89201074509445929287607067959,
0.89201074509445929287607067959, 2.04025144247440599431011115088, 2.422046696332539119013647260240, 3.3940051924703167013653131768, 4.01401067683220677870862320353, 4.90744494731475789598142654482, 5.62859583564405582061301903309, 6.20713541083947351938607876353, 6.904495287392644065696186137477, 7.29361255624994724676624978452, 8.58485465388722994339518214773, 9.39615972506210710887721056787, 9.950082838506073479573382514632, 10.49962611820311332142964372491, 11.57700729002383477774333791990, 12.04014671948622788740813324819, 12.75487746529069046323826835840, 13.38618182643212085511306068648, 13.95317736486632522114919534573, 14.41258776041154682294457508840, 15.27821527052289436983144672482, 15.87690844054797963448457424082, 16.59397104525626087049685641198, 17.198331100419834657516944168802, 17.70881198383319493853119635200