L(s) = 1 | − 5·2-s − 7·4-s + 94·5-s + 195·8-s − 470·10-s − 52·11-s + 770·13-s − 751·16-s − 2.02e3·17-s − 1.73e3·19-s − 658·20-s + 260·22-s + 576·23-s + 5.71e3·25-s − 3.85e3·26-s − 5.51e3·29-s − 6.33e3·31-s − 2.48e3·32-s + 1.01e4·34-s − 7.33e3·37-s + 8.66e3·38-s + 1.83e4·40-s − 3.26e3·41-s + 5.42e3·43-s + 364·44-s − 2.88e3·46-s + 864·47-s + ⋯ |
L(s) = 1 | − 0.883·2-s − 0.218·4-s + 1.68·5-s + 1.07·8-s − 1.48·10-s − 0.129·11-s + 1.26·13-s − 0.733·16-s − 1.69·17-s − 1.10·19-s − 0.367·20-s + 0.114·22-s + 0.227·23-s + 1.82·25-s − 1.11·26-s − 1.21·29-s − 1.18·31-s − 0.428·32-s + 1.49·34-s − 0.881·37-s + 0.972·38-s + 1.81·40-s − 0.303·41-s + 0.447·43-s + 0.0283·44-s − 0.200·46-s + 0.0570·47-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 | \( 1 + 5 T + p^{5} T^{2} \) |
| 5 | \( 1 - 94 T + p^{5} T^{2} \) |
| 11 | \( 1 + 52 T + p^{5} T^{2} \) |
| 13 | \( 1 - 770 T + p^{5} T^{2} \) |
| 17 | \( 1 + 2022 T + p^{5} T^{2} \) |
| 19 | \( 1 + 1732 T + p^{5} T^{2} \) |
| 23 | \( 1 - 576 T + p^{5} T^{2} \) |
| 29 | \( 1 + 5518 T + p^{5} T^{2} \) |
| 31 | \( 1 + 6336 T + p^{5} T^{2} \) |
| 37 | \( 1 + 7338 T + p^{5} T^{2} \) |
| 41 | \( 1 + 3262 T + p^{5} T^{2} \) |
| 43 | \( 1 - 5420 T + p^{5} T^{2} \) |
| 47 | \( 1 - 864 T + p^{5} T^{2} \) |
| 53 | \( 1 + 4182 T + p^{5} T^{2} \) |
| 59 | \( 1 + 11220 T + p^{5} T^{2} \) |
| 61 | \( 1 - 45602 T + p^{5} T^{2} \) |
| 67 | \( 1 - 1396 T + p^{5} T^{2} \) |
| 71 | \( 1 + 18720 T + p^{5} T^{2} \) |
| 73 | \( 1 + 46362 T + p^{5} T^{2} \) |
| 79 | \( 1 - 97424 T + p^{5} T^{2} \) |
| 83 | \( 1 + 81228 T + p^{5} T^{2} \) |
| 89 | \( 1 + 3182 T + p^{5} T^{2} \) |
| 97 | \( 1 + 4914 T + p^{5} T^{2} \) |
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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
−9.686879902953372506515996983686, −8.959289522120979125219220056616, −8.530524527063562654135362451645, −7.05665041472596775420207211064, −6.17916793218106919672180562051, −5.21700641444844076965709051319, −3.99393072620464901544562362410, −2.19698957695456137349275900442, −1.47948069039689251462319951770, 0,
1.47948069039689251462319951770, 2.19698957695456137349275900442, 3.99393072620464901544562362410, 5.21700641444844076965709051319, 6.17916793218106919672180562051, 7.05665041472596775420207211064, 8.530524527063562654135362451645, 8.959289522120979125219220056616, 9.686879902953372506515996983686