L(s) = 1 | − 64·2-s + 4.09e3·4-s − 2.62e5·8-s − 1.92e6·11-s + 1.67e7·16-s + 4.52e7·17-s − 8.79e7·19-s + 1.23e8·22-s + 2.44e8·25-s − 1.07e9·32-s − 2.89e9·34-s + 5.62e9·38-s − 8.62e9·41-s − 7.03e9·43-s − 7.87e9·44-s + 1.38e10·49-s − 1.56e10·50-s − 8.63e9·59-s + 6.87e10·64-s + 1.75e11·67-s + 1.85e11·68-s + 4.91e10·73-s − 3.60e11·76-s + 5.52e11·82-s + 1.92e11·83-s + 4.49e11·86-s + 5.04e11·88-s + ⋯ |
L(s) = 1 | − 2-s + 4-s − 8-s − 1.08·11-s + 16-s + 1.87·17-s − 1.86·19-s + 1.08·22-s + 25-s − 32-s − 1.87·34-s + 1.86·38-s − 1.81·41-s − 1.11·43-s − 1.08·44-s + 49-s − 50-s − 0.204·59-s + 64-s + 1.93·67-s + 1.87·68-s + 0.324·73-s − 1.86·76-s + 1.81·82-s + 0.590·83-s + 1.11·86-s + 1.08·88-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{13}{2})\) |
\(\approx\) |
\(1.032907053\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.032907053\) |
\(L(7)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + p^{6} T \) |
| 3 | \( 1 \) |
good | 5 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 7 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 11 | \( 1 + 1923122 T + p^{12} T^{2} \) |
| 13 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 17 | \( 1 - 45296062 T + p^{12} T^{2} \) |
| 19 | \( 1 + 87931438 T + p^{12} T^{2} \) |
| 23 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 29 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 31 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 37 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 41 | \( 1 + 8628259682 T + p^{12} T^{2} \) |
| 43 | \( 1 + 7030618702 T + p^{12} T^{2} \) |
| 47 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 53 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 59 | \( 1 + 8638314482 T + p^{12} T^{2} \) |
| 61 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 67 | \( 1 - 175045819538 T + p^{12} T^{2} \) |
| 71 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 73 | \( 1 - 49139489378 T + p^{12} T^{2} \) |
| 79 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 83 | \( 1 - 192940233262 T + p^{12} T^{2} \) |
| 89 | \( 1 - 866326445278 T + p^{12} T^{2} \) |
| 97 | \( 1 - 1656488134658 T + p^{12} T^{2} \) |
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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
−12.01327197133192754076936257447, −10.66782154489306705865091336129, −10.06674718532524411932992872016, −8.643758172068792181719992313412, −7.81596081253419646631743599396, −6.57398016214277117871036339438, −5.25394708590742550536800017901, −3.28164968065611900867587269377, −2.01842683137815890590778162772, −0.61014332829218763801154699599,
0.61014332829218763801154699599, 2.01842683137815890590778162772, 3.28164968065611900867587269377, 5.25394708590742550536800017901, 6.57398016214277117871036339438, 7.81596081253419646631743599396, 8.643758172068792181719992313412, 10.06674718532524411932992872016, 10.66782154489306705865091336129, 12.01327197133192754076936257447