| L(s) = 1 | + 5·5-s + 34·7-s − 18·11-s + 12·13-s + 106·17-s − 44·19-s − 56·23-s + 25·25-s − 270·29-s + 204·31-s + 170·35-s + 120·37-s − 80·41-s + 536·43-s + 536·47-s + 813·49-s − 542·53-s − 90·55-s + 174·59-s + 186·61-s + 60·65-s + 332·67-s + 132·71-s − 602·73-s − 612·77-s − 548·79-s + 492·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.83·7-s − 0.493·11-s + 0.256·13-s + 1.51·17-s − 0.531·19-s − 0.507·23-s + 1/5·25-s − 1.72·29-s + 1.18·31-s + 0.821·35-s + 0.533·37-s − 0.304·41-s + 1.90·43-s + 1.66·47-s + 2.37·49-s − 1.40·53-s − 0.220·55-s + 0.383·59-s + 0.390·61-s + 0.114·65-s + 0.605·67-s + 0.220·71-s − 0.965·73-s − 0.905·77-s − 0.780·79-s + 0.650·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.555245529\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.555245529\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - p T \) |
| good | 7 | \( 1 - 34 T + p^{3} T^{2} \) |
| 11 | \( 1 + 18 T + p^{3} T^{2} \) |
| 13 | \( 1 - 12 T + p^{3} T^{2} \) |
| 17 | \( 1 - 106 T + p^{3} T^{2} \) |
| 19 | \( 1 + 44 T + p^{3} T^{2} \) |
| 23 | \( 1 + 56 T + p^{3} T^{2} \) |
| 29 | \( 1 + 270 T + p^{3} T^{2} \) |
| 31 | \( 1 - 204 T + p^{3} T^{2} \) |
| 37 | \( 1 - 120 T + p^{3} T^{2} \) |
| 41 | \( 1 + 80 T + p^{3} T^{2} \) |
| 43 | \( 1 - 536 T + p^{3} T^{2} \) |
| 47 | \( 1 - 536 T + p^{3} T^{2} \) |
| 53 | \( 1 + 542 T + p^{3} T^{2} \) |
| 59 | \( 1 - 174 T + p^{3} T^{2} \) |
| 61 | \( 1 - 186 T + p^{3} T^{2} \) |
| 67 | \( 1 - 332 T + p^{3} T^{2} \) |
| 71 | \( 1 - 132 T + p^{3} T^{2} \) |
| 73 | \( 1 + 602 T + p^{3} T^{2} \) |
| 79 | \( 1 + 548 T + p^{3} T^{2} \) |
| 83 | \( 1 - 492 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1052 T + p^{3} T^{2} \) |
| 97 | \( 1 - 482 T + p^{3} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.98679851646944674611934443417, −10.25456747308672925354468931729, −9.111833886331446598961566694918, −8.040894913918139671838909358356, −7.55436782453444298124042213765, −5.92971449473199183222025263698, −5.18631822886258417681497016724, −4.06485044906358242201932983244, −2.36199022672682453869749096494, −1.19551817842177905216842005676,
1.19551817842177905216842005676, 2.36199022672682453869749096494, 4.06485044906358242201932983244, 5.18631822886258417681497016724, 5.92971449473199183222025263698, 7.55436782453444298124042213765, 8.040894913918139671838909358356, 9.111833886331446598961566694918, 10.25456747308672925354468931729, 10.98679851646944674611934443417
