L(s) = 1 | + (0.707 − 0.707i)3-s + (−0.224 − 2.22i)5-s + (3.14 + 3.14i)7-s − 1.00i·9-s − 1.41i·11-s + (−2.44 − 2.44i)13-s + (−1.73 − 1.41i)15-s + (4.44 − 4.44i)17-s + 0.635·19-s + 4.44·21-s + (−2.04 + 2.04i)23-s + (−4.89 + i)25-s + (−0.707 − 0.707i)27-s + 9.34i·29-s + 8.48i·31-s + ⋯ |
L(s) = 1 | + (0.408 − 0.408i)3-s + (−0.100 − 0.994i)5-s + (1.18 + 1.18i)7-s − 0.333i·9-s − 0.426i·11-s + (−0.679 − 0.679i)13-s + (−0.447 − 0.365i)15-s + (1.07 − 1.07i)17-s + 0.145·19-s + 0.970·21-s + (−0.427 + 0.427i)23-s + (−0.979 + 0.200i)25-s + (−0.136 − 0.136i)27-s + 1.73i·29-s + 1.52i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 + 0.655i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.755 + 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.39745 - 0.521778i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.39745 - 0.521778i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 + (0.224 + 2.22i)T \) |
good | 7 | \( 1 + (-3.14 - 3.14i)T + 7iT^{2} \) |
| 11 | \( 1 + 1.41iT - 11T^{2} \) |
| 13 | \( 1 + (2.44 + 2.44i)T + 13iT^{2} \) |
| 17 | \( 1 + (-4.44 + 4.44i)T - 17iT^{2} \) |
| 19 | \( 1 - 0.635T + 19T^{2} \) |
| 23 | \( 1 + (2.04 - 2.04i)T - 23iT^{2} \) |
| 29 | \( 1 - 9.34iT - 29T^{2} \) |
| 31 | \( 1 - 8.48iT - 31T^{2} \) |
| 37 | \( 1 + (1.55 - 1.55i)T - 37iT^{2} \) |
| 41 | \( 1 + 1.10T + 41T^{2} \) |
| 43 | \( 1 + (-0.635 + 0.635i)T - 43iT^{2} \) |
| 47 | \( 1 + (5.51 + 5.51i)T + 47iT^{2} \) |
| 53 | \( 1 + (2 + 2i)T + 53iT^{2} \) |
| 59 | \( 1 + 9.89T + 59T^{2} \) |
| 61 | \( 1 - 2.89T + 61T^{2} \) |
| 67 | \( 1 + (-6.29 - 6.29i)T + 67iT^{2} \) |
| 71 | \( 1 - 12.5iT - 71T^{2} \) |
| 73 | \( 1 + (-3 - 3i)T + 73iT^{2} \) |
| 79 | \( 1 + 9.75T + 79T^{2} \) |
| 83 | \( 1 + (-1.41 + 1.41i)T - 83iT^{2} \) |
| 89 | \( 1 + 2iT - 89T^{2} \) |
| 97 | \( 1 + (-3 + 3i)T - 97iT^{2} \) |
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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
−12.15428845150764390156830472063, −11.40037309222205723866332371863, −9.896503003125242626890691256140, −8.807660006706194385423232137884, −8.277684820066252016880554996352, −7.30789553522961502156117431552, −5.47401791763435416639709600300, −5.03144286266106297850108938134, −3.07657551701670098253775551160, −1.49839896983758720461681348397,
2.06823761243242344176880823045, 3.77666414638299488091156137477, 4.60063221497993749331731080442, 6.25132644272921907285217418742, 7.65714861364846184345796169038, 7.88002059106949841207212988015, 9.635450802630028945945385763678, 10.31621845858811008141961663756, 11.13120194669358276169269147170, 12.00826853064248297358948858295