| L(s) = 1 | + 9i·3-s + (−55 − 10i)5-s − 4i·7-s − 81·9-s + 500·11-s − 288i·13-s + (90 − 495i)15-s − 1.51e3i·17-s − 1.34e3·19-s + 36·21-s + 4.10e3i·23-s + (2.92e3 + 1.10e3i)25-s − 729i·27-s + 2.64e3·29-s + 5.61e3·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + (−0.983 − 0.178i)5-s − 0.0308i·7-s − 0.333·9-s + 1.24·11-s − 0.472i·13-s + (0.103 − 0.568i)15-s − 1.27i·17-s − 0.854·19-s + 0.0178·21-s + 1.61i·23-s + (0.936 + 0.352i)25-s − 0.192i·27-s + 0.584·29-s + 1.04·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.178 - 0.983i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.178 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.381325251\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.381325251\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 9iT \) |
| 5 | \( 1 + (55 + 10i)T \) |
| good | 7 | \( 1 + 4iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 500T + 1.61e5T^{2} \) |
| 13 | \( 1 + 288iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 1.51e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 1.34e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.10e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 2.64e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.61e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 7.28e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 1.89e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.40e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 8.90e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 3.98e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 2.83e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.82e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.59e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 2.88e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.08e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 6.02e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.46e3iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 2.26e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 3.69e4iT - 8.58e9T^{2} \) |
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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
−11.69782167442206144775883138861, −10.55941209883342872718618195504, −9.473144008617710720752269041042, −8.643658348492452052289983702555, −7.58730004858450567929147851818, −6.48634777799783507503717016141, −5.03796778033828196776290972658, −4.08579894411393318795105142425, −3.05244430074198062550540727309, −1.00743887055576110195755109867,
0.50279250293168393462702627084, 1.98549572103489739987571344828, 3.59843393810766853176200248235, 4.54012922107060818629922820693, 6.37020272512917050035432909804, 6.84204660766136714508617821745, 8.262108870725686046623952956376, 8.710312219705324126293446320646, 10.26010291015012095007904221638, 11.20747081541582643734793956714
