| L(s) = 1 | − 3·5-s + (2 − 1.73i)7-s + 5.19i·11-s − 3.46i·13-s + 6·17-s − 1.73i·19-s − 5.19i·23-s + 4·25-s − 10.3i·29-s − 5.19i·31-s + (−6 + 5.19i)35-s + 37-s + 3·41-s + 10·43-s + 6·47-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + (0.755 − 0.654i)7-s + 1.56i·11-s − 0.960i·13-s + 1.45·17-s − 0.397i·19-s − 1.08i·23-s + 0.800·25-s − 1.92i·29-s − 0.933i·31-s + (−1.01 + 0.878i)35-s + 0.164·37-s + 0.468·41-s + 1.52·43-s + 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 + 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.654 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.12965 - 0.516083i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.12965 - 0.516083i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2 + 1.73i)T \) |
| good | 5 | \( 1 + 3T + 5T^{2} \) |
| 11 | \( 1 - 5.19iT - 11T^{2} \) |
| 13 | \( 1 + 3.46iT - 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + 1.73iT - 19T^{2} \) |
| 23 | \( 1 + 5.19iT - 23T^{2} \) |
| 29 | \( 1 + 10.3iT - 29T^{2} \) |
| 31 | \( 1 + 5.19iT - 31T^{2} \) |
| 37 | \( 1 - T + 37T^{2} \) |
| 41 | \( 1 - 3T + 41T^{2} \) |
| 43 | \( 1 - 10T + 43T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 - 13.8iT - 61T^{2} \) |
| 67 | \( 1 - 2T + 67T^{2} \) |
| 71 | \( 1 + 5.19iT - 71T^{2} \) |
| 73 | \( 1 - 3.46iT - 73T^{2} \) |
| 79 | \( 1 - 14T + 79T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 + 9T + 89T^{2} \) |
| 97 | \( 1 - 6.92iT - 97T^{2} \) |
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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
−10.32662390470099114042626281047, −9.492523271290072175519272352245, −8.044381298194251625091481078884, −7.79811924854856239466814564671, −7.10425072879305392438658208391, −5.67158657434365688064545595987, −4.45338876063546516600267605152, −4.04746434361632659910029501738, −2.55425424424961638468067319166, −0.75941191870987508666180872782,
1.28279118866956139487243444151, 3.12451173773084013834398890970, 3.86416325348673531727637195774, 5.10732015917325321888303997167, 5.90273728814475041929538409788, 7.22069651647561925637587961467, 7.961556206862797752154927028510, 8.610287254908329453313484316998, 9.387314706235804928673910862984, 10.89805773830269221677894138933
