L(s) = 1 | − 3.84·5-s − 2.14·7-s − 0.389i·11-s − 5.51i·13-s − 6.90·17-s − 6.97·19-s − 4.97·23-s + 9.78·25-s − 7.02i·29-s + 0.566·31-s + 8.25·35-s + 7.77i·37-s − 8.87·41-s − 3.56i·43-s − 12.8i·47-s + ⋯ |
L(s) = 1 | − 1.71·5-s − 0.811·7-s − 0.117i·11-s − 1.52i·13-s − 1.67·17-s − 1.59·19-s − 1.03·23-s + 1.95·25-s − 1.30i·29-s + 0.101·31-s + 1.39·35-s + 1.27i·37-s − 1.38·41-s − 0.544i·43-s − 1.86i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.606 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.606 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.02145335527\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.02145335527\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 167 | \( 1 + (-3.86 - 12.3i)T \) |
good | 5 | \( 1 + 3.84T + 5T^{2} \) |
| 7 | \( 1 + 2.14T + 7T^{2} \) |
| 11 | \( 1 + 0.389iT - 11T^{2} \) |
| 13 | \( 1 + 5.51iT - 13T^{2} \) |
| 17 | \( 1 + 6.90T + 17T^{2} \) |
| 19 | \( 1 + 6.97T + 19T^{2} \) |
| 23 | \( 1 + 4.97T + 23T^{2} \) |
| 29 | \( 1 + 7.02iT - 29T^{2} \) |
| 31 | \( 1 - 0.566T + 31T^{2} \) |
| 37 | \( 1 - 7.77iT - 37T^{2} \) |
| 41 | \( 1 + 8.87T + 41T^{2} \) |
| 43 | \( 1 + 3.56iT - 43T^{2} \) |
| 47 | \( 1 + 12.8iT - 47T^{2} \) |
| 53 | \( 1 + 1.41T + 53T^{2} \) |
| 59 | \( 1 + 7.04T + 59T^{2} \) |
| 61 | \( 1 - 2.28T + 61T^{2} \) |
| 67 | \( 1 + 12.4iT - 67T^{2} \) |
| 71 | \( 1 + 2.35T + 71T^{2} \) |
| 73 | \( 1 + 13.7iT - 73T^{2} \) |
| 79 | \( 1 - 6.60iT - 79T^{2} \) |
| 83 | \( 1 - 10.0T + 83T^{2} \) |
| 89 | \( 1 - 13.1iT - 89T^{2} \) |
| 97 | \( 1 + 13.0T + 97T^{2} \) |
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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
−7.64501463281675814847578886691, −6.53723323132869053360360790031, −6.46790126997157454261236413091, −5.17697243207856816410528552211, −4.37052861237135553356290689748, −3.78452454805353681750364441766, −3.10847712423457821365313942874, −2.14591292312197475680469181381, −0.30154698691227104481253567227, −0.01744623214716556780844733827,
1.74360845544807508308421926573, 2.74393107494041061438243185610, 3.76435200391774732618738472303, 4.26561085697560262456035970816, 4.68556882140697306114950093482, 6.11539905483958893328992775590, 6.75396453427274119558809446455, 7.08833948171089315666325457996, 8.022568773929599490504407592075, 8.727260424623320337430878792581