| L(s) = 1 | − 2·2-s − 4·3-s − 28·4-s + 25·5-s + 8·6-s − 192·7-s + 120·8-s − 227·9-s − 50·10-s + 112·12-s − 286·13-s + 384·14-s − 100·15-s + 656·16-s + 1.67e3·17-s + 454·18-s − 1.06e3·19-s − 700·20-s + 768·21-s + 2.97e3·23-s − 480·24-s + 625·25-s + 572·26-s + 1.88e3·27-s + 5.37e3·28-s + 3.41e3·29-s + 200·30-s + ⋯ |
| L(s) = 1 | − 0.353·2-s − 0.256·3-s − 7/8·4-s + 0.447·5-s + 0.0907·6-s − 1.48·7-s + 0.662·8-s − 0.934·9-s − 0.158·10-s + 0.224·12-s − 0.469·13-s + 0.523·14-s − 0.114·15-s + 0.640·16-s + 1.40·17-s + 0.330·18-s − 0.673·19-s − 0.391·20-s + 0.380·21-s + 1.17·23-s − 0.170·24-s + 1/5·25-s + 0.165·26-s + 0.496·27-s + 1.29·28-s + 0.752·29-s + 0.0405·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - p^{2} T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + p T + p^{5} T^{2} \) |
| 3 | \( 1 + 4 T + p^{5} T^{2} \) |
| 7 | \( 1 + 192 T + p^{5} T^{2} \) |
| 13 | \( 1 + 22 p T + p^{5} T^{2} \) |
| 17 | \( 1 - 1678 T + p^{5} T^{2} \) |
| 19 | \( 1 + 1060 T + p^{5} T^{2} \) |
| 23 | \( 1 - 2976 T + p^{5} T^{2} \) |
| 29 | \( 1 - 3410 T + p^{5} T^{2} \) |
| 31 | \( 1 + 2448 T + p^{5} T^{2} \) |
| 37 | \( 1 - 182 T + p^{5} T^{2} \) |
| 41 | \( 1 - 9398 T + p^{5} T^{2} \) |
| 43 | \( 1 - 1244 T + p^{5} T^{2} \) |
| 47 | \( 1 + 12088 T + p^{5} T^{2} \) |
| 53 | \( 1 - 23846 T + p^{5} T^{2} \) |
| 59 | \( 1 + 20020 T + p^{5} T^{2} \) |
| 61 | \( 1 + 32302 T + p^{5} T^{2} \) |
| 67 | \( 1 - 60972 T + p^{5} T^{2} \) |
| 71 | \( 1 + 32648 T + p^{5} T^{2} \) |
| 73 | \( 1 - 38774 T + p^{5} T^{2} \) |
| 79 | \( 1 - 33360 T + p^{5} T^{2} \) |
| 83 | \( 1 + 16716 T + p^{5} T^{2} \) |
| 89 | \( 1 - 101370 T + p^{5} T^{2} \) |
| 97 | \( 1 + 119038 T + p^{5} T^{2} \) |
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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
−9.445025877096281863912867103310, −8.818047051011451094762642450646, −7.78539686155843011902013236999, −6.64722377028941179743740404052, −5.77114612761005024280352272066, −4.97096708387848048165979181405, −3.60062406471790934374519601766, −2.72087006683386035959102187036, −0.943843379484922464814806595377, 0,
0.943843379484922464814806595377, 2.72087006683386035959102187036, 3.60062406471790934374519601766, 4.97096708387848048165979181405, 5.77114612761005024280352272066, 6.64722377028941179743740404052, 7.78539686155843011902013236999, 8.818047051011451094762642450646, 9.445025877096281863912867103310
