| L(s) = 1 | + 2·3-s + 7-s + 9-s − 6·11-s + 4·13-s + 2·17-s − 4·19-s + 2·21-s + 23-s − 4·27-s − 10·29-s + 8·31-s − 12·33-s + 8·37-s + 8·39-s − 2·41-s + 6·43-s + 12·47-s + 49-s + 4·51-s − 12·53-s − 8·57-s + 6·59-s − 6·61-s + 63-s − 2·67-s + 2·69-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.377·7-s + 1/3·9-s − 1.80·11-s + 1.10·13-s + 0.485·17-s − 0.917·19-s + 0.436·21-s + 0.208·23-s − 0.769·27-s − 1.85·29-s + 1.43·31-s − 2.08·33-s + 1.31·37-s + 1.28·39-s − 0.312·41-s + 0.914·43-s + 1.75·47-s + 1/7·49-s + 0.560·51-s − 1.64·53-s − 1.05·57-s + 0.781·59-s − 0.768·61-s + 0.125·63-s − 0.244·67-s + 0.240·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.032806638\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.032806638\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.22352456293199, −13.67506911726391, −13.27142873026874, −12.98040350002019, −12.40993750099642, −11.63646832782870, −10.99602855445644, −10.79677155554340, −10.12522378584050, −9.591696127029613, −8.956266988863860, −8.517089039619124, −8.115963398586659, −7.528272872906612, −7.385571398908840, −6.154277346468957, −5.908685507163184, −5.253751800915623, −4.467044454023256, −4.015833396155552, −3.219124437037027, −2.770053408180352, −2.224087384680488, −1.534027794017740, −0.5260700166563104,
0.5260700166563104, 1.534027794017740, 2.224087384680488, 2.770053408180352, 3.219124437037027, 4.015833396155552, 4.467044454023256, 5.253751800915623, 5.908685507163184, 6.154277346468957, 7.385571398908840, 7.528272872906612, 8.115963398586659, 8.517089039619124, 8.956266988863860, 9.591696127029613, 10.12522378584050, 10.79677155554340, 10.99602855445644, 11.63646832782870, 12.40993750099642, 12.98040350002019, 13.27142873026874, 13.67506911726391, 14.22352456293199
