L(s) = 1 | − i·2-s − 4-s − 5-s + i·8-s + i·10-s − 1.37i·11-s − 4.02i·13-s + 16-s + 0.648·17-s + 5.77i·19-s + 20-s − 1.37·22-s + 1.66i·23-s + 25-s − 4.02·26-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.5·4-s − 0.447·5-s + 0.353i·8-s + 0.316i·10-s − 0.415i·11-s − 1.11i·13-s + 0.250·16-s + 0.157·17-s + 1.32i·19-s + 0.223·20-s − 0.293·22-s + 0.347i·23-s + 0.200·25-s − 0.789·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 - 0.192i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.981 - 0.192i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6973390235\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6973390235\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 1.37iT - 11T^{2} \) |
| 13 | \( 1 + 4.02iT - 13T^{2} \) |
| 17 | \( 1 - 0.648T + 17T^{2} \) |
| 19 | \( 1 - 5.77iT - 19T^{2} \) |
| 23 | \( 1 - 1.66iT - 23T^{2} \) |
| 29 | \( 1 + 2.53iT - 29T^{2} \) |
| 31 | \( 1 + 6.20iT - 31T^{2} \) |
| 37 | \( 1 - 10.9T + 37T^{2} \) |
| 41 | \( 1 + 6.26T + 41T^{2} \) |
| 43 | \( 1 + 7.21T + 43T^{2} \) |
| 47 | \( 1 - 0.598T + 47T^{2} \) |
| 53 | \( 1 + 7.48iT - 53T^{2} \) |
| 59 | \( 1 + 3.71T + 59T^{2} \) |
| 61 | \( 1 + 4.94iT - 61T^{2} \) |
| 67 | \( 1 + 3.76T + 67T^{2} \) |
| 71 | \( 1 + 3.97iT - 71T^{2} \) |
| 73 | \( 1 - 7.53iT - 73T^{2} \) |
| 79 | \( 1 - 10.4T + 79T^{2} \) |
| 83 | \( 1 + 4.08T + 83T^{2} \) |
| 89 | \( 1 + 9.76T + 89T^{2} \) |
| 97 | \( 1 + 7.65iT - 97T^{2} \) |
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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
−8.033777168594372220193214496012, −7.57799773455937788624804585306, −6.35071186925663791596270218770, −5.70127810608432939684790150348, −4.91751689259876932428224175440, −3.92649037553479881062734642078, −3.38615556690760943040105359276, −2.49220999874239394329600734031, −1.33598819011788357003698648238, −0.21468498454927292898444342637,
1.25673658990841010466043816249, 2.54555638740655439977892807077, 3.54454564786844757099034849862, 4.56760977487572867690661040962, 4.82701533891871930217477906809, 5.94846419075428602894983629909, 6.79167261133863016136804954514, 7.10503246363770847058130471858, 7.957499679371861085334339146797, 8.725328365079836742055815086234