| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4261 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4261 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.299375414\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.299375414\) |
| \(L(1)\) |
\(\approx\) |
\(1.269752817\) |
| \(L(1)\) |
\(\approx\) |
\(1.269752817\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 4261 | \( 1 \) |
| 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
−18.71030731557061803126037951468, −17.649337126006822990368005748091, −17.169228602306959384095286369512, −16.31903856932639664496265961318, −15.7494974624103097986588262542, −15.15866374533406797836341493542, −14.264805865605238035226857115994, −13.62114803932629504417032632269, −13.052960080691563960483972286907, −12.30219724731649819473073660660, −11.27817000167283217306480366967, −10.60362301446713347029872841583, −9.73735464135971730582538473435, −9.37561403656290462209369933577, −8.9075140656880925419105542631, −8.24410023009923597827626039153, −7.13642630698811646382464808571, −6.64995903516883535917059700792, −6.16298466026178178527502205914, −5.05897429329944146019092294993, −3.7165061494279591357089996426, −3.2313528400270290348710463646, −2.41031724964753752560924417645, −1.61416764599173953479743232994, −0.95633025422685285321980275275,
0.95633025422685285321980275275, 1.61416764599173953479743232994, 2.41031724964753752560924417645, 3.2313528400270290348710463646, 3.7165061494279591357089996426, 5.05897429329944146019092294993, 6.16298466026178178527502205914, 6.64995903516883535917059700792, 7.13642630698811646382464808571, 8.24410023009923597827626039153, 8.9075140656880925419105542631, 9.37561403656290462209369933577, 9.73735464135971730582538473435, 10.60362301446713347029872841583, 11.27817000167283217306480366967, 12.30219724731649819473073660660, 13.052960080691563960483972286907, 13.62114803932629504417032632269, 14.264805865605238035226857115994, 15.15866374533406797836341493542, 15.7494974624103097986588262542, 16.31903856932639664496265961318, 17.169228602306959384095286369512, 17.649337126006822990368005748091, 18.71030731557061803126037951468
