| L(s) = 1 | + 9·3-s − 25·5-s − 44·7-s + 81·9-s − 216·11-s + 770·13-s − 225·15-s + 534·17-s − 1.58e3·19-s − 396·21-s − 2.90e3·23-s + 625·25-s + 729·27-s − 4.56e3·29-s − 2.74e3·31-s − 1.94e3·33-s + 1.10e3·35-s + 1.44e3·37-s + 6.93e3·39-s − 1.33e4·41-s − 1.72e4·43-s − 2.02e3·45-s + 1.08e4·47-s − 1.48e4·49-s + 4.80e3·51-s − 9.94e3·53-s + 5.40e3·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.339·7-s + 1/3·9-s − 0.538·11-s + 1.26·13-s − 0.258·15-s + 0.448·17-s − 1.00·19-s − 0.195·21-s − 1.14·23-s + 1/5·25-s + 0.192·27-s − 1.00·29-s − 0.512·31-s − 0.310·33-s + 0.151·35-s + 0.173·37-s + 0.729·39-s − 1.24·41-s − 1.41·43-s − 0.149·45-s + 0.714·47-s − 0.884·49-s + 0.258·51-s − 0.486·53-s + 0.240·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p^{2} T \) |
| 5 | \( 1 + p^{2} T \) |
| good | 7 | \( 1 + 44 T + p^{5} T^{2} \) |
| 11 | \( 1 + 216 T + p^{5} T^{2} \) |
| 13 | \( 1 - 770 T + p^{5} T^{2} \) |
| 17 | \( 1 - 534 T + p^{5} T^{2} \) |
| 19 | \( 1 + 1580 T + p^{5} T^{2} \) |
| 23 | \( 1 + 2904 T + p^{5} T^{2} \) |
| 29 | \( 1 + 4566 T + p^{5} T^{2} \) |
| 31 | \( 1 + 2744 T + p^{5} T^{2} \) |
| 37 | \( 1 - 1442 T + p^{5} T^{2} \) |
| 41 | \( 1 + 13350 T + p^{5} T^{2} \) |
| 43 | \( 1 + 17204 T + p^{5} T^{2} \) |
| 47 | \( 1 - 10824 T + p^{5} T^{2} \) |
| 53 | \( 1 + 9942 T + p^{5} T^{2} \) |
| 59 | \( 1 - 264 p T + p^{5} T^{2} \) |
| 61 | \( 1 - 39302 T + p^{5} T^{2} \) |
| 67 | \( 1 + 55796 T + p^{5} T^{2} \) |
| 71 | \( 1 + 57120 T + p^{5} T^{2} \) |
| 73 | \( 1 - 50402 T + p^{5} T^{2} \) |
| 79 | \( 1 - 10552 T + p^{5} T^{2} \) |
| 83 | \( 1 + 1308 p T + p^{5} T^{2} \) |
| 89 | \( 1 + 116430 T + p^{5} T^{2} \) |
| 97 | \( 1 + 2782 T + p^{5} T^{2} \) |
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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
−10.72550219251549467157388590693, −9.828540285695818883019962992709, −8.632651810133201404249153057468, −8.017119639937434573969098195076, −6.82428206816225941387665715357, −5.67633796325383228117443312381, −4.14607542532039733172956572392, −3.24776272207911159068350629043, −1.75302877067768597824223342507, 0,
1.75302877067768597824223342507, 3.24776272207911159068350629043, 4.14607542532039733172956572392, 5.67633796325383228117443312381, 6.82428206816225941387665715357, 8.017119639937434573969098195076, 8.632651810133201404249153057468, 9.828540285695818883019962992709, 10.72550219251549467157388590693
