| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.464646920\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.464646920\) |
| \(L(\frac{7}{2})\) |
|
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
−14.08538134992053622467889434153, −12.86486601535879112508894247606, −11.63024470210671134402691599109, −10.95736431160759327719729724129, −9.426608670009701488918531069469, −8.045201196000500845384061248828, −6.66990966001777794226367462003, −5.23752951386603078751630055725, −3.64115446092692935300086297823, −1.10615106348577456516118953071,
1.10615106348577456516118953071, 3.64115446092692935300086297823, 5.23752951386603078751630055725, 6.66990966001777794226367462003, 8.045201196000500845384061248828, 9.426608670009701488918531069469, 10.95736431160759327719729724129, 11.63024470210671134402691599109, 12.86486601535879112508894247606, 14.08538134992053622467889434153
