| L(s) = 1 | + 13·2-s + 41·4-s − 1.13e3·8-s − 8.68e3·11-s − 1.99e4·16-s − 1.12e5·22-s + 6.79e4·23-s − 7.81e4·25-s + 2.53e5·29-s − 1.14e5·32-s − 2.78e5·37-s + 1.01e6·43-s − 3.56e5·44-s + 8.83e5·46-s − 1.01e6·50-s + 1.80e6·53-s + 3.29e6·58-s + 1.06e6·64-s − 4.86e6·67-s + 3.72e6·71-s − 3.61e6·74-s + 1.10e6·79-s + 1.31e7·86-s + 9.82e6·88-s + 2.78e6·92-s − 3.20e6·100-s + 2.34e7·106-s + ⋯ |
| L(s) = 1 | + 1.14·2-s + 0.320·4-s − 0.780·8-s − 1.96·11-s − 1.21·16-s − 2.26·22-s + 1.16·23-s − 25-s + 1.93·29-s − 0.618·32-s − 0.903·37-s + 1.94·43-s − 0.630·44-s + 1.33·46-s − 1.14·50-s + 1.66·53-s + 2.21·58-s + 0.507·64-s − 1.97·67-s + 1.23·71-s − 1.03·74-s + 0.251·79-s + 2.23·86-s + 1.53·88-s + 0.373·92-s − 0.320·100-s + 1.91·106-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.558565250\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.558565250\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 13 T + p^{7} T^{2} \) |
| 5 | \( 1 + p^{7} T^{2} \) |
| 11 | \( 1 + 8684 T + p^{7} T^{2} \) |
| 13 | \( 1 + p^{7} T^{2} \) |
| 17 | \( 1 + p^{7} T^{2} \) |
| 19 | \( 1 + p^{7} T^{2} \) |
| 23 | \( 1 - 67976 T + p^{7} T^{2} \) |
| 29 | \( 1 - 253622 T + p^{7} T^{2} \) |
| 31 | \( 1 + p^{7} T^{2} \) |
| 37 | \( 1 + 278382 T + p^{7} T^{2} \) |
| 41 | \( 1 + p^{7} T^{2} \) |
| 43 | \( 1 - 1014204 T + p^{7} T^{2} \) |
| 47 | \( 1 + p^{7} T^{2} \) |
| 53 | \( 1 - 1804610 T + p^{7} T^{2} \) |
| 59 | \( 1 + p^{7} T^{2} \) |
| 61 | \( 1 + p^{7} T^{2} \) |
| 67 | \( 1 + 4863092 T + p^{7} T^{2} \) |
| 71 | \( 1 - 3721744 T + p^{7} T^{2} \) |
| 73 | \( 1 + p^{7} T^{2} \) |
| 79 | \( 1 - 1101752 T + p^{7} T^{2} \) |
| 83 | \( 1 + p^{7} T^{2} \) |
| 89 | \( 1 + p^{7} T^{2} \) |
| 97 | \( 1 + p^{7} T^{2} \) |
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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
−10.16972765026967418655404581066, −8.998309069090050057332974452507, −8.062021156148131218928536325516, −7.03508609341211168256493279929, −5.84471966023470115057532436114, −5.16594049101961122328847312795, −4.34448408500040583091128109046, −3.10167318798838540823912895228, −2.40291990361673072569603650144, −0.57918062571625166496922105297,
0.57918062571625166496922105297, 2.40291990361673072569603650144, 3.10167318798838540823912895228, 4.34448408500040583091128109046, 5.16594049101961122328847312795, 5.84471966023470115057532436114, 7.03508609341211168256493279929, 8.062021156148131218928536325516, 8.998309069090050057332974452507, 10.16972765026967418655404581066
