| L(s) = 1 | − 9·3-s − 34·5-s − 240·7-s + 81·9-s − 124·11-s + 46·13-s + 306·15-s + 1.95e3·17-s − 1.92e3·19-s + 2.16e3·21-s + 2.84e3·23-s − 1.96e3·25-s − 729·27-s − 8.92e3·29-s − 4.64e3·31-s + 1.11e3·33-s + 8.16e3·35-s − 4.36e3·37-s − 414·39-s − 2.88e3·41-s + 1.13e4·43-s − 2.75e3·45-s + 7.00e3·47-s + 4.07e4·49-s − 1.75e4·51-s − 2.25e4·53-s + 4.21e3·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.608·5-s − 1.85·7-s + 1/3·9-s − 0.308·11-s + 0.0754·13-s + 0.351·15-s + 1.63·17-s − 1.22·19-s + 1.06·21-s + 1.11·23-s − 0.630·25-s − 0.192·27-s − 1.97·29-s − 0.868·31-s + 0.178·33-s + 1.12·35-s − 0.523·37-s − 0.0435·39-s − 0.268·41-s + 0.934·43-s − 0.202·45-s + 0.462·47-s + 2.42·49-s − 0.946·51-s − 1.10·53-s + 0.187·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p^{2} T \) |
| good | 5 | \( 1 + 34 T + p^{5} T^{2} \) |
| 7 | \( 1 + 240 T + p^{5} T^{2} \) |
| 11 | \( 1 + 124 T + p^{5} T^{2} \) |
| 13 | \( 1 - 46 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1954 T + p^{5} T^{2} \) |
| 19 | \( 1 + 1924 T + p^{5} T^{2} \) |
| 23 | \( 1 - 2840 T + p^{5} T^{2} \) |
| 29 | \( 1 + 8922 T + p^{5} T^{2} \) |
| 31 | \( 1 + 4648 T + p^{5} T^{2} \) |
| 37 | \( 1 + 4362 T + p^{5} T^{2} \) |
| 41 | \( 1 + 2886 T + p^{5} T^{2} \) |
| 43 | \( 1 - 11332 T + p^{5} T^{2} \) |
| 47 | \( 1 - 7008 T + p^{5} T^{2} \) |
| 53 | \( 1 + 22594 T + p^{5} T^{2} \) |
| 59 | \( 1 + 28 T + p^{5} T^{2} \) |
| 61 | \( 1 + 6386 T + p^{5} T^{2} \) |
| 67 | \( 1 + 39076 T + p^{5} T^{2} \) |
| 71 | \( 1 + 54872 T + p^{5} T^{2} \) |
| 73 | \( 1 - 21034 T + p^{5} T^{2} \) |
| 79 | \( 1 - 26632 T + p^{5} T^{2} \) |
| 83 | \( 1 - 56188 T + p^{5} T^{2} \) |
| 89 | \( 1 - 64410 T + p^{5} T^{2} \) |
| 97 | \( 1 + 116158 T + p^{5} T^{2} \) |
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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
−16.21691457831029212101342686342, −15.07363701288664687994535817310, −13.11822900063559532739319403892, −12.30361318861663291966122988016, −10.70934464025058145094887144061, −9.400697991094110531193838053161, −7.34525751220318135605049171434, −5.85821873812974401161436880367, −3.55424539237826313179614857718, 0,
3.55424539237826313179614857718, 5.85821873812974401161436880367, 7.34525751220318135605049171434, 9.400697991094110531193838053161, 10.70934464025058145094887144061, 12.30361318861663291966122988016, 13.11822900063559532739319403892, 15.07363701288664687994535817310, 16.21691457831029212101342686342
