L(s) = 1 | + (0.565 + 1.29i)2-s + (1.40 − 1.01i)3-s + (−1.35 + 1.46i)4-s + 0.662i·5-s + (2.11 + 1.24i)6-s + 4.60i·7-s + (−2.67 − 0.933i)8-s + (0.932 − 2.85i)9-s + (−0.859 + 0.375i)10-s + 1.50·11-s + (−0.416 + 3.43i)12-s + 1.18·13-s + (−5.96 + 2.60i)14-s + (0.673 + 0.929i)15-s + (−0.301 − 3.98i)16-s − 1.95i·17-s + ⋯ |
L(s) = 1 | + (0.400 + 0.916i)2-s + (0.809 − 0.586i)3-s + (−0.679 + 0.733i)4-s + 0.296i·5-s + (0.861 + 0.507i)6-s + 1.73i·7-s + (−0.944 − 0.329i)8-s + (0.310 − 0.950i)9-s + (−0.271 + 0.118i)10-s + 0.452·11-s + (−0.120 + 0.992i)12-s + 0.328·13-s + (−1.59 + 0.695i)14-s + (0.174 + 0.240i)15-s + (−0.0752 − 0.997i)16-s − 0.473i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.120 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.120 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.37523 + 1.21882i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.37523 + 1.21882i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.565 - 1.29i)T \) |
| 3 | \( 1 + (-1.40 + 1.01i)T \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 - 0.662iT - 5T^{2} \) |
| 7 | \( 1 - 4.60iT - 7T^{2} \) |
| 11 | \( 1 - 1.50T + 11T^{2} \) |
| 13 | \( 1 - 1.18T + 13T^{2} \) |
| 17 | \( 1 + 1.95iT - 17T^{2} \) |
| 19 | \( 1 - 0.0109iT - 19T^{2} \) |
| 29 | \( 1 + 8.65iT - 29T^{2} \) |
| 31 | \( 1 - 6.06iT - 31T^{2} \) |
| 37 | \( 1 + 7.04T + 37T^{2} \) |
| 41 | \( 1 + 7.37iT - 41T^{2} \) |
| 43 | \( 1 + 4.67iT - 43T^{2} \) |
| 47 | \( 1 + 10.0T + 47T^{2} \) |
| 53 | \( 1 + 0.825iT - 53T^{2} \) |
| 59 | \( 1 - 6.63T + 59T^{2} \) |
| 61 | \( 1 - 12.1T + 61T^{2} \) |
| 67 | \( 1 - 0.812iT - 67T^{2} \) |
| 71 | \( 1 - 8.94T + 71T^{2} \) |
| 73 | \( 1 + 13.3T + 73T^{2} \) |
| 79 | \( 1 - 9.43iT - 79T^{2} \) |
| 83 | \( 1 - 1.34T + 83T^{2} \) |
| 89 | \( 1 - 6.30iT - 89T^{2} \) |
| 97 | \( 1 - 16.5T + 97T^{2} \) |
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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
−12.27422567324991405351575034201, −11.68438056703293800419240636163, −9.739644277831152576728332774346, −8.784291433988117004220582798312, −8.380533161765351727670286979638, −7.07788120654300456433247498186, −6.28538112450589249713410741362, −5.22320527754243992230274862341, −3.58478783947041897228023138881, −2.42701218574792894241598234067,
1.42672955132390386471448767754, 3.30184250805725569581587923961, 4.08484786086872483039659555991, 4.98021719677431372250291584376, 6.72453146007455523885868994686, 8.065365311454567879802041645129, 9.039524664126477130661604595246, 10.02261156288163300446618989760, 10.61266326121410757817521627932, 11.44350781122386515320008283191