| L(s) = 1 | + 9·3-s + 74·5-s − 49·7-s + 81·9-s − 216·11-s − 186·13-s + 666·15-s + 1.07e3·17-s + 908·19-s − 441·21-s + 2.23e3·23-s + 2.35e3·25-s + 729·27-s + 5.36e3·29-s + 536·31-s − 1.94e3·33-s − 3.62e3·35-s + 3.79e3·37-s − 1.67e3·39-s + 1.85e4·41-s − 1.53e4·43-s + 5.99e3·45-s − 2.34e4·47-s + 2.40e3·49-s + 9.70e3·51-s + 9.06e3·53-s − 1.59e4·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.32·5-s − 0.377·7-s + 1/3·9-s − 0.538·11-s − 0.305·13-s + 0.764·15-s + 0.904·17-s + 0.577·19-s − 0.218·21-s + 0.881·23-s + 0.752·25-s + 0.192·27-s + 1.18·29-s + 0.100·31-s − 0.310·33-s − 0.500·35-s + 0.456·37-s − 0.176·39-s + 1.72·41-s − 1.26·43-s + 0.441·45-s − 1.55·47-s + 1/7·49-s + 0.522·51-s + 0.443·53-s − 0.712·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.393488731\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.393488731\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{2} T \) |
| 7 | \( 1 + p^{2} T \) |
| good | 5 | \( 1 - 74 T + p^{5} T^{2} \) |
| 11 | \( 1 + 216 T + p^{5} T^{2} \) |
| 13 | \( 1 + 186 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1078 T + p^{5} T^{2} \) |
| 19 | \( 1 - 908 T + p^{5} T^{2} \) |
| 23 | \( 1 - 2236 T + p^{5} T^{2} \) |
| 29 | \( 1 - 5366 T + p^{5} T^{2} \) |
| 31 | \( 1 - 536 T + p^{5} T^{2} \) |
| 37 | \( 1 - 3798 T + p^{5} T^{2} \) |
| 41 | \( 1 - 18598 T + p^{5} T^{2} \) |
| 43 | \( 1 + 356 p T + p^{5} T^{2} \) |
| 47 | \( 1 + 23480 T + p^{5} T^{2} \) |
| 53 | \( 1 - 9062 T + p^{5} T^{2} \) |
| 59 | \( 1 - 49284 T + p^{5} T^{2} \) |
| 61 | \( 1 - 17806 T + p^{5} T^{2} \) |
| 67 | \( 1 + 24876 T + p^{5} T^{2} \) |
| 71 | \( 1 + 3468 T + p^{5} T^{2} \) |
| 73 | \( 1 + 32414 T + p^{5} T^{2} \) |
| 79 | \( 1 + 25384 T + p^{5} T^{2} \) |
| 83 | \( 1 - 67284 T + p^{5} T^{2} \) |
| 89 | \( 1 + 698 T + p^{5} T^{2} \) |
| 97 | \( 1 - 154906 T + p^{5} T^{2} \) |
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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
−10.29676846994767264201029595168, −9.867919411169775005879026368196, −9.021497900317909410353593460193, −7.936332326422677614900112969666, −6.86165681181230908469762973957, −5.81259329706953614323174337088, −4.87654840102774249509522996104, −3.23457328948938750230434637145, −2.34970707746789371560219178982, −1.03803521015984428248180962239,
1.03803521015984428248180962239, 2.34970707746789371560219178982, 3.23457328948938750230434637145, 4.87654840102774249509522996104, 5.81259329706953614323174337088, 6.86165681181230908469762973957, 7.936332326422677614900112969666, 9.021497900317909410353593460193, 9.867919411169775005879026368196, 10.29676846994767264201029595168
