| L(s) = 1 | + 5·3-s + 2·7-s − 2·9-s + 39·11-s + 84·13-s − 61·17-s + 151·19-s + 10·21-s − 58·23-s − 145·27-s + 192·29-s − 18·31-s + 195·33-s − 138·37-s + 420·39-s + 229·41-s − 164·43-s − 212·47-s − 339·49-s − 305·51-s + 578·53-s + 755·57-s − 336·59-s + 858·61-s − 4·63-s − 209·67-s − 290·69-s + ⋯ |
| L(s) = 1 | + 0.962·3-s + 0.107·7-s − 0.0740·9-s + 1.06·11-s + 1.79·13-s − 0.870·17-s + 1.82·19-s + 0.103·21-s − 0.525·23-s − 1.03·27-s + 1.22·29-s − 0.104·31-s + 1.02·33-s − 0.613·37-s + 1.72·39-s + 0.872·41-s − 0.581·43-s − 0.657·47-s − 0.988·49-s − 0.837·51-s + 1.49·53-s + 1.75·57-s − 0.741·59-s + 1.80·61-s − 0.00799·63-s − 0.381·67-s − 0.505·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.595673122\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.595673122\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 5 T + p^{3} T^{2} \) |
| 7 | \( 1 - 2 T + p^{3} T^{2} \) |
| 11 | \( 1 - 39 T + p^{3} T^{2} \) |
| 13 | \( 1 - 84 T + p^{3} T^{2} \) |
| 17 | \( 1 + 61 T + p^{3} T^{2} \) |
| 19 | \( 1 - 151 T + p^{3} T^{2} \) |
| 23 | \( 1 + 58 T + p^{3} T^{2} \) |
| 29 | \( 1 - 192 T + p^{3} T^{2} \) |
| 31 | \( 1 + 18 T + p^{3} T^{2} \) |
| 37 | \( 1 + 138 T + p^{3} T^{2} \) |
| 41 | \( 1 - 229 T + p^{3} T^{2} \) |
| 43 | \( 1 + 164 T + p^{3} T^{2} \) |
| 47 | \( 1 + 212 T + p^{3} T^{2} \) |
| 53 | \( 1 - 578 T + p^{3} T^{2} \) |
| 59 | \( 1 + 336 T + p^{3} T^{2} \) |
| 61 | \( 1 - 858 T + p^{3} T^{2} \) |
| 67 | \( 1 + 209 T + p^{3} T^{2} \) |
| 71 | \( 1 + 780 T + p^{3} T^{2} \) |
| 73 | \( 1 + 403 T + p^{3} T^{2} \) |
| 79 | \( 1 + 230 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1293 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1369 T + p^{3} T^{2} \) |
| 97 | \( 1 - 382 T + p^{3} T^{2} \) |
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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
−11.85340200077995382974986278357, −11.18557493684478897875068215159, −9.807469515722484982055208540286, −8.842760142606351954510600134901, −8.277835662904540116112287939789, −6.92577140210464799514359797394, −5.78253706349857504577230947665, −4.07965010062562905854019559774, −3.08039223659535500209288017213, −1.39244343621259215093755400948,
1.39244343621259215093755400948, 3.08039223659535500209288017213, 4.07965010062562905854019559774, 5.78253706349857504577230947665, 6.92577140210464799514359797394, 8.277835662904540116112287939789, 8.842760142606351954510600134901, 9.807469515722484982055208540286, 11.18557493684478897875068215159, 11.85340200077995382974986278357
