L(s) = 1 | + (14.8 + 4.61i)3-s + (31.2 + 46.3i)5-s + (−91.3 + 91.3i)7-s + (200. + 137. i)9-s + 99.2i·11-s + (−452. − 452. i)13-s + (252. + 834. i)15-s + (1.26e3 + 1.26e3i)17-s + 912. i·19-s + (−1.78e3 + 939. i)21-s + (1.74e3 − 1.74e3i)23-s + (−1.16e3 + 2.89e3i)25-s + (2.35e3 + 2.97e3i)27-s + 299.·29-s + 2.59e3·31-s + ⋯ |
L(s) = 1 | + (0.955 + 0.295i)3-s + (0.559 + 0.828i)5-s + (−0.705 + 0.705i)7-s + (0.824 + 0.565i)9-s + 0.247i·11-s + (−0.742 − 0.742i)13-s + (0.289 + 0.957i)15-s + (1.06 + 1.06i)17-s + 0.580i·19-s + (−0.882 + 0.464i)21-s + (0.688 − 0.688i)23-s + (−0.373 + 0.927i)25-s + (0.620 + 0.784i)27-s + 0.0661·29-s + 0.484·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.256 - 0.966i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.256 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.83777 + 1.41315i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.83777 + 1.41315i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-14.8 - 4.61i)T \) |
| 5 | \( 1 + (-31.2 - 46.3i)T \) |
good | 7 | \( 1 + (91.3 - 91.3i)T - 1.68e4iT^{2} \) |
| 11 | \( 1 - 99.2iT - 1.61e5T^{2} \) |
| 13 | \( 1 + (452. + 452. i)T + 3.71e5iT^{2} \) |
| 17 | \( 1 + (-1.26e3 - 1.26e3i)T + 1.41e6iT^{2} \) |
| 19 | \( 1 - 912. iT - 2.47e6T^{2} \) |
| 23 | \( 1 + (-1.74e3 + 1.74e3i)T - 6.43e6iT^{2} \) |
| 29 | \( 1 - 299.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.59e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + (-5.91e3 + 5.91e3i)T - 6.93e7iT^{2} \) |
| 41 | \( 1 + 1.94e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 + (8.60e3 + 8.60e3i)T + 1.47e8iT^{2} \) |
| 47 | \( 1 + (1.73e4 + 1.73e4i)T + 2.29e8iT^{2} \) |
| 53 | \( 1 + (-1.94e4 + 1.94e4i)T - 4.18e8iT^{2} \) |
| 59 | \( 1 - 220.T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.39e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + (3.86e4 - 3.86e4i)T - 1.35e9iT^{2} \) |
| 71 | \( 1 - 6.18e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + (1.40e4 + 1.40e4i)T + 2.07e9iT^{2} \) |
| 79 | \( 1 + 6.48e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 + (-5.90e3 + 5.90e3i)T - 3.93e9iT^{2} \) |
| 89 | \( 1 - 1.25e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + (7.06e4 - 7.06e4i)T - 8.58e9iT^{2} \) |
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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
−14.67808629969956234261603389158, −13.27347313075061303431197007143, −12.33450936978251578031447033068, −10.38740374768155903701331698780, −9.834756940432595583556395057476, −8.484439688846264006400908360581, −7.11086111050712711046538308421, −5.59558342614273312284451785246, −3.46418589006882639448598991381, −2.29388131653503489712154546357,
1.07941628758719541280200705579, 2.96965198017690813920923352732, 4.71471717803099341975768934110, 6.64413803350933444984712726506, 7.86676324822937589361303426642, 9.346135532997099164335483463158, 9.830291479889281815858731539652, 11.84192559448442714097256113278, 13.09829226148207187004486358105, 13.65868166645406787512290490014