L(s) = 1 | + (6.66 − 6.66i)3-s + (3.32 − 3.32i)5-s + (−12.8 − 12.8i)7-s − 61.7i·9-s + (30.6 + 30.6i)11-s − 42.1·13-s − 44.2i·15-s + (52.4 − 46.5i)17-s + 112. i·19-s − 171.·21-s + (−49.9 − 49.9i)23-s + 102. i·25-s + (−231. − 231. i)27-s + (134. − 134. i)29-s + (75.2 − 75.2i)31-s + ⋯ |
L(s) = 1 | + (1.28 − 1.28i)3-s + (0.297 − 0.297i)5-s + (−0.693 − 0.693i)7-s − 2.28i·9-s + (0.841 + 0.841i)11-s − 0.900·13-s − 0.761i·15-s + (0.747 − 0.663i)17-s + 1.35i·19-s − 1.77·21-s + (−0.452 − 0.452i)23-s + 0.823i·25-s + (−1.65 − 1.65i)27-s + (0.862 − 0.862i)29-s + (0.435 − 0.435i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.181 + 0.983i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.181 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.52585 - 1.83253i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.52585 - 1.83253i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 17 | \( 1 + (-52.4 + 46.5i)T \) |
good | 3 | \( 1 + (-6.66 + 6.66i)T - 27iT^{2} \) |
| 5 | \( 1 + (-3.32 + 3.32i)T - 125iT^{2} \) |
| 7 | \( 1 + (12.8 + 12.8i)T + 343iT^{2} \) |
| 11 | \( 1 + (-30.6 - 30.6i)T + 1.33e3iT^{2} \) |
| 13 | \( 1 + 42.1T + 2.19e3T^{2} \) |
| 19 | \( 1 - 112. iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (49.9 + 49.9i)T + 1.21e4iT^{2} \) |
| 29 | \( 1 + (-134. + 134. i)T - 2.43e4iT^{2} \) |
| 31 | \( 1 + (-75.2 + 75.2i)T - 2.97e4iT^{2} \) |
| 37 | \( 1 + (-246. + 246. i)T - 5.06e4iT^{2} \) |
| 41 | \( 1 + (-170. - 170. i)T + 6.89e4iT^{2} \) |
| 43 | \( 1 - 112. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 188.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 642. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 431. iT - 2.05e5T^{2} \) |
| 61 | \( 1 + (41.0 + 41.0i)T + 2.26e5iT^{2} \) |
| 67 | \( 1 - 173.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (232. - 232. i)T - 3.57e5iT^{2} \) |
| 73 | \( 1 + (197. - 197. i)T - 3.89e5iT^{2} \) |
| 79 | \( 1 + (-936. - 936. i)T + 4.93e5iT^{2} \) |
| 83 | \( 1 - 168. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 1.10e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-1.20e3 + 1.20e3i)T - 9.12e5iT^{2} \) |
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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
−12.56978540648662001257851211748, −12.01736527352078966944397400389, −9.868838128574571707947990532697, −9.400890172044165868103748506831, −7.948426030195185963452112487533, −7.27233480578363082469675963052, −6.21704494862749629114431227554, −4.03273805617324873154415317055, −2.58346040917408728412301194428, −1.15161978656404496336489005028,
2.58979619819620027243442167496, 3.48741780355681909089024958885, 4.91426045225014540049169974058, 6.42579701438807851002475874341, 8.118465540068987840661139754136, 9.058080030499228807570945185654, 9.721521264220116145669377380134, 10.60592765265596217541313731746, 11.98022739515787699012428739523, 13.34240010631776353231385749065