| L(s) = 1 | − 62·2-s − 243·3-s + 1.79e3·4-s + 3.31e3·5-s + 1.50e4·6-s + 1.56e4·8-s + 5.90e4·9-s − 2.05e5·10-s + 6.28e5·11-s − 4.36e5·12-s − 1.76e5·13-s − 8.04e5·15-s − 4.64e6·16-s − 5.66e5·17-s − 3.66e6·18-s + 1.29e7·19-s + 5.94e6·20-s − 3.89e7·22-s − 2.56e7·23-s − 3.79e6·24-s − 3.78e7·25-s + 1.09e7·26-s − 1.43e7·27-s − 4.74e7·29-s + 4.98e7·30-s − 1.39e7·31-s + 2.56e8·32-s + ⋯ |
| L(s) = 1 | − 1.37·2-s − 0.577·3-s + 0.876·4-s + 0.473·5-s + 0.790·6-s + 0.168·8-s + 1/3·9-s − 0.648·10-s + 1.17·11-s − 0.506·12-s − 0.132·13-s − 0.273·15-s − 1.10·16-s − 0.0968·17-s − 0.456·18-s + 1.19·19-s + 0.415·20-s − 1.61·22-s − 0.831·23-s − 0.0973·24-s − 0.775·25-s + 0.180·26-s − 0.192·27-s − 0.429·29-s + 0.374·30-s − 0.0874·31-s + 1.34·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + p^{5} T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 31 p T + p^{11} T^{2} \) |
| 5 | \( 1 - 662 p T + p^{11} T^{2} \) |
| 11 | \( 1 - 628904 T + p^{11} T^{2} \) |
| 13 | \( 1 + 176854 T + p^{11} T^{2} \) |
| 17 | \( 1 + 566958 T + p^{11} T^{2} \) |
| 19 | \( 1 - 679796 p T + p^{11} T^{2} \) |
| 23 | \( 1 + 25664100 T + p^{11} T^{2} \) |
| 29 | \( 1 + 47411458 T + p^{11} T^{2} \) |
| 31 | \( 1 + 13942680 T + p^{11} T^{2} \) |
| 37 | \( 1 + 641657298 T + p^{11} T^{2} \) |
| 41 | \( 1 - 600859298 T + p^{11} T^{2} \) |
| 43 | \( 1 + 1417753612 T + p^{11} T^{2} \) |
| 47 | \( 1 + 860414040 T + p^{11} T^{2} \) |
| 53 | \( 1 - 3221420478 T + p^{11} T^{2} \) |
| 59 | \( 1 - 6082959012 T + p^{11} T^{2} \) |
| 61 | \( 1 - 864141122 T + p^{11} T^{2} \) |
| 67 | \( 1 - 11897667268 T + p^{11} T^{2} \) |
| 71 | \( 1 + 14077803900 T + p^{11} T^{2} \) |
| 73 | \( 1 - 18814150398 T + p^{11} T^{2} \) |
| 79 | \( 1 - 17021361416 T + p^{11} T^{2} \) |
| 83 | \( 1 + 47613135564 T + p^{11} T^{2} \) |
| 89 | \( 1 + 61562070254 T + p^{11} T^{2} \) |
| 97 | \( 1 - 166479510534 T + p^{11} T^{2} \) |
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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
−10.11118902159821347784917991281, −9.594420979519080666970442576824, −8.601476884095934408009048382138, −7.42082398413152601746059761709, −6.51284284495850755002204306751, −5.29769585403786351498257191951, −3.84672887276195041779688073666, −1.97329193674056451069041618941, −1.12823241072364512353802983044, 0,
1.12823241072364512353802983044, 1.97329193674056451069041618941, 3.84672887276195041779688073666, 5.29769585403786351498257191951, 6.51284284495850755002204306751, 7.42082398413152601746059761709, 8.601476884095934408009048382138, 9.594420979519080666970442576824, 10.11118902159821347784917991281
