| L(s) = 1 | + (−1.57 + 2.55i)3-s + 1.31i·5-s − 10.2·7-s + (−4.01 − 8.05i)9-s − 16.6i·11-s + 18.7·13-s + (−3.35 − 2.07i)15-s − 4.38i·17-s + 11.5·19-s + (16.1 − 26.1i)21-s + 16.7i·23-s + 23.2·25-s + (26.8 + 2.46i)27-s + 12.5i·29-s − 20.3·31-s + ⋯ |
| L(s) = 1 | + (−0.526 + 0.850i)3-s + 0.263i·5-s − 1.46·7-s + (−0.446 − 0.894i)9-s − 1.51i·11-s + 1.44·13-s + (−0.223 − 0.138i)15-s − 0.257i·17-s + 0.608·19-s + (0.769 − 1.24i)21-s + 0.728i·23-s + 0.930·25-s + (0.995 + 0.0912i)27-s + 0.432i·29-s − 0.655·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.526i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.850 + 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.951897 - 0.270665i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.951897 - 0.270665i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.57 - 2.55i)T \) |
| good | 5 | \( 1 - 1.31iT - 25T^{2} \) |
| 7 | \( 1 + 10.2T + 49T^{2} \) |
| 11 | \( 1 + 16.6iT - 121T^{2} \) |
| 13 | \( 1 - 18.7T + 169T^{2} \) |
| 17 | \( 1 + 4.38iT - 289T^{2} \) |
| 19 | \( 1 - 11.5T + 361T^{2} \) |
| 23 | \( 1 - 16.7iT - 529T^{2} \) |
| 29 | \( 1 - 12.5iT - 841T^{2} \) |
| 31 | \( 1 + 20.3T + 961T^{2} \) |
| 37 | \( 1 + 18.5T + 1.36e3T^{2} \) |
| 41 | \( 1 + 78.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 36.4T + 1.84e3T^{2} \) |
| 47 | \( 1 + 19.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 81.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 29.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 72.0T + 3.72e3T^{2} \) |
| 67 | \( 1 - 56.3T + 4.48e3T^{2} \) |
| 71 | \( 1 + 136. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 80.8T + 5.32e3T^{2} \) |
| 79 | \( 1 + 86.0T + 6.24e3T^{2} \) |
| 83 | \( 1 + 80.4iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 131. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 20.4T + 9.40e3T^{2} \) |
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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
−10.94465330432287389318633637142, −10.27896002279338177162270061582, −9.200510555636234671281163620030, −8.665369355269045341633405314249, −6.97081325847389003210229237950, −6.08705978256123349581151783304, −5.42655032358615403474507116527, −3.58681651553584042806156545902, −3.32437891016868888822833385207, −0.55036363486887212782945184508,
1.18776845487793353497849622326, 2.77373104944893571037160484838, 4.26346289360657038071459306105, 5.64423585903623719660018869940, 6.52800684440847176081238032070, 7.18301646467573892167962444446, 8.376026040463315345563030443672, 9.422763381927462918408093508814, 10.31876798200679797823084920694, 11.28414596709055785211403997896
