L(s) = 1 | + (−2.51 + 2.51i)5-s − 7-s + (0.984 + 0.984i)11-s + (−3.26 + 3.26i)13-s − 5.50i·17-s + (1.21 + 1.21i)19-s − 8.62i·23-s − 7.65i·25-s + (−2.04 − 2.04i)29-s + 0.164i·31-s + (2.51 − 2.51i)35-s + (−8.31 − 8.31i)37-s + 9.56·41-s + (−5.01 + 5.01i)43-s − 3.37·47-s + ⋯ |
L(s) = 1 | + (−1.12 + 1.12i)5-s − 0.377·7-s + (0.296 + 0.296i)11-s + (−0.904 + 0.904i)13-s − 1.33i·17-s + (0.278 + 0.278i)19-s − 1.79i·23-s − 1.53i·25-s + (−0.380 − 0.380i)29-s + 0.0295i·31-s + (0.425 − 0.425i)35-s + (−1.36 − 1.36i)37-s + 1.49·41-s + (−0.765 + 0.765i)43-s − 0.492·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0912i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9595091146\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9595091146\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 + (2.51 - 2.51i)T - 5iT^{2} \) |
| 11 | \( 1 + (-0.984 - 0.984i)T + 11iT^{2} \) |
| 13 | \( 1 + (3.26 - 3.26i)T - 13iT^{2} \) |
| 17 | \( 1 + 5.50iT - 17T^{2} \) |
| 19 | \( 1 + (-1.21 - 1.21i)T + 19iT^{2} \) |
| 23 | \( 1 + 8.62iT - 23T^{2} \) |
| 29 | \( 1 + (2.04 + 2.04i)T + 29iT^{2} \) |
| 31 | \( 1 - 0.164iT - 31T^{2} \) |
| 37 | \( 1 + (8.31 + 8.31i)T + 37iT^{2} \) |
| 41 | \( 1 - 9.56T + 41T^{2} \) |
| 43 | \( 1 + (5.01 - 5.01i)T - 43iT^{2} \) |
| 47 | \( 1 + 3.37T + 47T^{2} \) |
| 53 | \( 1 + (-3.72 + 3.72i)T - 53iT^{2} \) |
| 59 | \( 1 + (-10.0 - 10.0i)T + 59iT^{2} \) |
| 61 | \( 1 + (1.07 - 1.07i)T - 61iT^{2} \) |
| 67 | \( 1 + (-3.12 - 3.12i)T + 67iT^{2} \) |
| 71 | \( 1 - 7.66iT - 71T^{2} \) |
| 73 | \( 1 - 8.40iT - 73T^{2} \) |
| 79 | \( 1 - 13.8iT - 79T^{2} \) |
| 83 | \( 1 + (-7.31 + 7.31i)T - 83iT^{2} \) |
| 89 | \( 1 - 7.49T + 89T^{2} \) |
| 97 | \( 1 + 7.84T + 97T^{2} \) |
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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
−8.349760778751844500000435501823, −7.44435111263935172160276064137, −7.01446687494802954327283001192, −6.59610150513786353268855322194, −5.43056735185383425382643006482, −4.41678427571817977146032824010, −3.90371860490421914531526873977, −2.87557658648741989242745502437, −2.28166548333722834532368320405, −0.43831887739740624210913634665,
0.68323412823179568740150700108, 1.80041340095874601332272479623, 3.37579107299227699935773614497, 3.65941025516154919889701480642, 4.79094319026546946612136800699, 5.29595151495772096913341098825, 6.17474620044298629220170267025, 7.23185004587352536945146766982, 7.78628220561142225443211625797, 8.393867705502385091922240554633