| L(s) = 1 | − 10·2-s + 68·4-s − 106·5-s − 360·8-s + 1.06e3·10-s − 92·11-s − 670·13-s + 1.42e3·16-s − 222·17-s + 908·19-s − 7.20e3·20-s + 920·22-s + 1.17e3·23-s + 8.11e3·25-s + 6.70e3·26-s − 1.11e3·29-s − 3.69e3·31-s − 2.72e3·32-s + 2.22e3·34-s + 4.18e3·37-s − 9.08e3·38-s + 3.81e4·40-s − 6.66e3·41-s − 3.70e3·43-s − 6.25e3·44-s − 1.17e4·46-s − 7.05e3·47-s + ⋯ |
| L(s) = 1 | − 1.76·2-s + 17/8·4-s − 1.89·5-s − 1.98·8-s + 3.35·10-s − 0.229·11-s − 1.09·13-s + 1.39·16-s − 0.186·17-s + 0.577·19-s − 4.02·20-s + 0.405·22-s + 0.463·23-s + 2.59·25-s + 1.94·26-s − 0.246·29-s − 0.690·31-s − 0.469·32-s + 0.329·34-s + 0.502·37-s − 1.02·38-s + 3.77·40-s − 0.618·41-s − 0.305·43-s − 0.487·44-s − 0.819·46-s − 0.465·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 5 p T + p^{5} T^{2} \) |
| 5 | \( 1 + 106 T + p^{5} T^{2} \) |
| 11 | \( 1 + 92 T + p^{5} T^{2} \) |
| 13 | \( 1 + 670 T + p^{5} T^{2} \) |
| 17 | \( 1 + 222 T + p^{5} T^{2} \) |
| 19 | \( 1 - 908 T + p^{5} T^{2} \) |
| 23 | \( 1 - 1176 T + p^{5} T^{2} \) |
| 29 | \( 1 + 1118 T + p^{5} T^{2} \) |
| 31 | \( 1 + 3696 T + p^{5} T^{2} \) |
| 37 | \( 1 - 4182 T + p^{5} T^{2} \) |
| 41 | \( 1 + 6662 T + p^{5} T^{2} \) |
| 43 | \( 1 + 3700 T + p^{5} T^{2} \) |
| 47 | \( 1 + 7056 T + p^{5} T^{2} \) |
| 53 | \( 1 - 37578 T + p^{5} T^{2} \) |
| 59 | \( 1 - 32700 T + p^{5} T^{2} \) |
| 61 | \( 1 - 10802 T + p^{5} T^{2} \) |
| 67 | \( 1 - 64996 T + p^{5} T^{2} \) |
| 71 | \( 1 - 61320 T + p^{5} T^{2} \) |
| 73 | \( 1 + 38922 T + p^{5} T^{2} \) |
| 79 | \( 1 + 88096 T + p^{5} T^{2} \) |
| 83 | \( 1 - 71892 T + p^{5} T^{2} \) |
| 89 | \( 1 - 111818 T + p^{5} T^{2} \) |
| 97 | \( 1 - 150846 T + p^{5} T^{2} \) |
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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
−9.775865617062240615917375954378, −8.807599811440872421700104775814, −8.084397123135757461402854255721, −7.38277825396877907234190714424, −6.87129327667452542188194726089, −5.03708361954040388251443282166, −3.66930635584257412832888041876, −2.45747446749185591119437553658, −0.839612222336772734132484303529, 0,
0.839612222336772734132484303529, 2.45747446749185591119437553658, 3.66930635584257412832888041876, 5.03708361954040388251443282166, 6.87129327667452542188194726089, 7.38277825396877907234190714424, 8.084397123135757461402854255721, 8.807599811440872421700104775814, 9.775865617062240615917375954378
