| L(s) = 1 | − 218·7-s + 480·11-s + 622·13-s + 186·17-s − 1.20e3·19-s − 3.18e3·23-s − 5.52e3·29-s + 9.35e3·31-s − 5.61e3·37-s + 1.43e4·41-s + 370·43-s + 1.61e4·47-s + 3.07e4·49-s − 4.37e3·53-s + 1.17e4·59-s + 1.32e4·61-s + 1.15e4·67-s + 2.95e4·71-s − 3.36e4·73-s − 1.04e5·77-s + 3.12e4·79-s − 3.84e4·83-s − 1.19e5·89-s − 1.35e5·91-s − 9.46e4·97-s − 1.01e5·101-s + 1.43e5·103-s + ⋯ |
| L(s) = 1 | − 1.68·7-s + 1.19·11-s + 1.02·13-s + 0.156·17-s − 0.765·19-s − 1.25·23-s − 1.22·29-s + 1.74·31-s − 0.674·37-s + 1.33·41-s + 0.0305·43-s + 1.06·47-s + 1.82·49-s − 0.213·53-s + 0.439·59-s + 0.454·61-s + 0.314·67-s + 0.695·71-s − 0.740·73-s − 2.01·77-s + 0.562·79-s − 0.612·83-s − 1.59·89-s − 1.71·91-s − 1.02·97-s − 0.985·101-s + 1.33·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 218 T + p^{5} T^{2} \) |
| 11 | \( 1 - 480 T + p^{5} T^{2} \) |
| 13 | \( 1 - 622 T + p^{5} T^{2} \) |
| 17 | \( 1 - 186 T + p^{5} T^{2} \) |
| 19 | \( 1 + 1204 T + p^{5} T^{2} \) |
| 23 | \( 1 + 3186 T + p^{5} T^{2} \) |
| 29 | \( 1 + 5526 T + p^{5} T^{2} \) |
| 31 | \( 1 - 9356 T + p^{5} T^{2} \) |
| 37 | \( 1 + 5618 T + p^{5} T^{2} \) |
| 41 | \( 1 - 14394 T + p^{5} T^{2} \) |
| 43 | \( 1 - 370 T + p^{5} T^{2} \) |
| 47 | \( 1 - 16146 T + p^{5} T^{2} \) |
| 53 | \( 1 + 4374 T + p^{5} T^{2} \) |
| 59 | \( 1 - 11748 T + p^{5} T^{2} \) |
| 61 | \( 1 - 13202 T + p^{5} T^{2} \) |
| 67 | \( 1 - 11542 T + p^{5} T^{2} \) |
| 71 | \( 1 - 29532 T + p^{5} T^{2} \) |
| 73 | \( 1 + 33698 T + p^{5} T^{2} \) |
| 79 | \( 1 - 31208 T + p^{5} T^{2} \) |
| 83 | \( 1 + 38466 T + p^{5} T^{2} \) |
| 89 | \( 1 + 119514 T + p^{5} T^{2} \) |
| 97 | \( 1 + 94658 T + p^{5} T^{2} \) |
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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
−9.075003587638005920331822676629, −8.240031626356498227548558167673, −7.01478990035479629313222952316, −6.30318208016214102507362891792, −5.82424060610866342464953835681, −4.11309507038530784198285152956, −3.66998092761753360253919505163, −2.49365786329364883076661292258, −1.14424231893348904837975705067, 0,
1.14424231893348904837975705067, 2.49365786329364883076661292258, 3.66998092761753360253919505163, 4.11309507038530784198285152956, 5.82424060610866342464953835681, 6.30318208016214102507362891792, 7.01478990035479629313222952316, 8.240031626356498227548558167673, 9.075003587638005920331822676629
