L(s) = 1 | − 3.36·5-s + (−2.57 − 0.595i)7-s + 1.64i·11-s − 3.06i·13-s + 3.36·17-s − 5.53i·19-s + 1.64i·23-s + 6.29·25-s − 6.06i·29-s − 4.33i·31-s + (8.66 + 2i)35-s − 7.29·37-s − 0.979·41-s − 5.15·43-s + 11.1·47-s + ⋯ |
L(s) = 1 | − 1.50·5-s + (−0.974 − 0.224i)7-s + 0.496i·11-s − 0.851i·13-s + 0.814·17-s − 1.26i·19-s + 0.343i·23-s + 1.25·25-s − 1.12i·29-s − 0.779i·31-s + (1.46 + 0.338i)35-s − 1.19·37-s − 0.152·41-s − 0.786·43-s + 1.63·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.378 - 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.378 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2319975675\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2319975675\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.57 + 0.595i)T \) |
good | 5 | \( 1 + 3.36T + 5T^{2} \) |
| 11 | \( 1 - 1.64iT - 11T^{2} \) |
| 13 | \( 1 + 3.06iT - 13T^{2} \) |
| 17 | \( 1 - 3.36T + 17T^{2} \) |
| 19 | \( 1 + 5.53iT - 19T^{2} \) |
| 23 | \( 1 - 1.64iT - 23T^{2} \) |
| 29 | \( 1 + 6.06iT - 29T^{2} \) |
| 31 | \( 1 + 4.33iT - 31T^{2} \) |
| 37 | \( 1 + 7.29T + 37T^{2} \) |
| 41 | \( 1 + 0.979T + 41T^{2} \) |
| 43 | \( 1 + 5.15T + 43T^{2} \) |
| 47 | \( 1 - 11.1T + 47T^{2} \) |
| 53 | \( 1 + 4.24iT - 53T^{2} \) |
| 59 | \( 1 + 6.13T + 59T^{2} \) |
| 61 | \( 1 - 11.1iT - 61T^{2} \) |
| 67 | \( 1 + 8.48T + 67T^{2} \) |
| 71 | \( 1 - 5.64iT - 71T^{2} \) |
| 73 | \( 1 + 8.11iT - 73T^{2} \) |
| 79 | \( 1 + 8.48T + 79T^{2} \) |
| 83 | \( 1 - 11.1T + 83T^{2} \) |
| 89 | \( 1 + 7.70T + 89T^{2} \) |
| 97 | \( 1 + 14.2iT - 97T^{2} \) |
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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
−8.640754973401002163308598488372, −7.61682361882600091228799794958, −7.50213165025066073253556482975, −6.61980841990857108335905870731, −5.71706001649462647748787741538, −4.79527355353556460203639565954, −3.96715267673931356136045132144, −3.36964032881398007003710708294, −2.55386220618203789129927309718, −0.817109307966961458869510831114,
0.094754581607637965195365630208, 1.50304451541976637594583479291, 3.03396136865196719432989858335, 3.53020400266749539554140861017, 4.19654698382446642950490385744, 5.21824688849871589435135525684, 6.06827618177583271450693897846, 6.89481276911101898972307286274, 7.41929492631661915227545724055, 8.296536766299438351311025258035