L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 7-s − 8-s + 9-s + 10-s + 11-s + 12-s + 13-s + 14-s − 15-s + 16-s + 17-s − 18-s + 19-s − 20-s − 21-s − 22-s + 23-s − 24-s + 25-s − 26-s + 27-s − 28-s + ⋯ |
L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 7-s − 8-s + 9-s + 10-s + 11-s + 12-s + 13-s + 14-s − 15-s + 16-s + 17-s − 18-s + 19-s − 20-s − 21-s − 22-s + 23-s − 24-s + 25-s − 26-s + 27-s − 28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2677 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2677 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.491156312\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.491156312\) |
\(L(1)\) |
\(\approx\) |
\(0.9551675129\) |
\(L(1)\) |
\(\approx\) |
\(0.9551675129\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2677 | \( 1 \) |
good | 2 | \( 1 - T \) |
| 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 | \( 1 + T \) |
| 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
−19.31973139797284455590545661233, −18.79667014019903466813454273403, −18.240667123588314229822438933656, −17.08832798967373688256832623752, −16.279741229345555747936537644311, −15.93362573851790893500845410379, −15.267568996594608348739118551893, −14.50813221038910608171394653337, −13.74969366858794794828522984452, −12.646547785366333210199618117614, −12.192113440844774743338758310799, −11.30332199258924736213991567488, −10.53007383138464996775547627649, −9.633200765501926321769748791004, −9.171917601648937537499351689, −8.45046961751257119827751254371, −7.843009008734283653730780161353, −6.9158911441271819107900614863, −6.66243431273016138341668627314, −5.360144923986247177556655306057, −3.924555872416612211305778593974, −3.362873643165082525881181754445, −2.94052953270084430614924025337, −1.5177675442132531984409599813, −0.866568269596737446360548640199,
0.866568269596737446360548640199, 1.5177675442132531984409599813, 2.94052953270084430614924025337, 3.362873643165082525881181754445, 3.924555872416612211305778593974, 5.360144923986247177556655306057, 6.66243431273016138341668627314, 6.9158911441271819107900614863, 7.843009008734283653730780161353, 8.45046961751257119827751254371, 9.171917601648937537499351689, 9.633200765501926321769748791004, 10.53007383138464996775547627649, 11.30332199258924736213991567488, 12.192113440844774743338758310799, 12.646547785366333210199618117614, 13.74969366858794794828522984452, 14.50813221038910608171394653337, 15.267568996594608348739118551893, 15.93362573851790893500845410379, 16.279741229345555747936537644311, 17.08832798967373688256832623752, 18.240667123588314229822438933656, 18.79667014019903466813454273403, 19.31973139797284455590545661233