| L(s) = 1 | + 5.54·3-s − 25·5-s − 49·7-s − 212.·9-s + 713.·11-s − 157.·13-s − 138.·15-s − 1.66e3·17-s + 18.8·19-s − 271.·21-s + 1.47e3·23-s + 625·25-s − 2.52e3·27-s + 7.55e3·29-s + 6.29e3·31-s + 3.95e3·33-s + 1.22e3·35-s − 391.·37-s − 873.·39-s − 5.92e3·41-s + 2.00e4·43-s + 5.30e3·45-s − 2.03e4·47-s + 2.40e3·49-s − 9.25e3·51-s + 2.08e3·53-s − 1.78e4·55-s + ⋯ |
| L(s) = 1 | + 0.355·3-s − 0.447·5-s − 0.377·7-s − 0.873·9-s + 1.77·11-s − 0.258·13-s − 0.159·15-s − 1.40·17-s + 0.0119·19-s − 0.134·21-s + 0.583·23-s + 0.200·25-s − 0.666·27-s + 1.66·29-s + 1.17·31-s + 0.632·33-s + 0.169·35-s − 0.0470·37-s − 0.0919·39-s − 0.550·41-s + 1.65·43-s + 0.390·45-s − 1.34·47-s + 0.142·49-s − 0.498·51-s + 0.102·53-s − 0.795·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.904162004\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.904162004\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 25T \) |
| 7 | \( 1 + 49T \) |
| good | 3 | \( 1 - 5.54T + 243T^{2} \) |
| 11 | \( 1 - 713.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 157.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.66e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 18.8T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.47e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 7.55e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.29e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 391.T + 6.93e7T^{2} \) |
| 41 | \( 1 + 5.92e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.00e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.03e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.08e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.50e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.17e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.42e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.63e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.54e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 8.45e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.85e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.32e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.39e5T + 8.58e9T^{2} \) |
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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
−11.27726225938089305309898380101, −9.928090440939386275510040728145, −8.922490988274632964412540815794, −8.403893074851653906774878035890, −6.92998822530646856574997155987, −6.28290161101631193290815822826, −4.69218487704418856627163516373, −3.64406156055255957449885518535, −2.46992631114777564647962591854, −0.77490516035717051737954373031,
0.77490516035717051737954373031, 2.46992631114777564647962591854, 3.64406156055255957449885518535, 4.69218487704418856627163516373, 6.28290161101631193290815822826, 6.92998822530646856574997155987, 8.403893074851653906774878035890, 8.922490988274632964412540815794, 9.928090440939386275510040728145, 11.27726225938089305309898380101
