L(s) = 1 | − 2.28i·3-s − 2.23·5-s − 2.82i·7-s − 2.23·9-s − 5.23·11-s + 4.03i·13-s + 5.11i·15-s + 1.08i·17-s + 19-s − 6.47·21-s − 7.40i·23-s + 5.00·25-s − 1.74i·27-s − 4.47·29-s − 4·31-s + ⋯ |
L(s) = 1 | − 1.32i·3-s − 0.999·5-s − 1.06i·7-s − 0.745·9-s − 1.57·11-s + 1.11i·13-s + 1.32i·15-s + 0.262i·17-s + 0.229·19-s − 1.41·21-s − 1.54i·23-s + 1.00·25-s − 0.336i·27-s − 0.830·29-s − 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(-0.649728i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.649728i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + 2.23T \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 + 2.28iT - 3T^{2} \) |
| 7 | \( 1 + 2.82iT - 7T^{2} \) |
| 11 | \( 1 + 5.23T + 11T^{2} \) |
| 13 | \( 1 - 4.03iT - 13T^{2} \) |
| 17 | \( 1 - 1.08iT - 17T^{2} \) |
| 23 | \( 1 + 7.40iT - 23T^{2} \) |
| 29 | \( 1 + 4.47T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 6.86iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 8.48iT - 43T^{2} \) |
| 47 | \( 1 + 8.48iT - 47T^{2} \) |
| 53 | \( 1 + 6.86iT - 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 - 1.70T + 61T^{2} \) |
| 67 | \( 1 + 1.62iT - 67T^{2} \) |
| 71 | \( 1 + 1.52T + 71T^{2} \) |
| 73 | \( 1 + 13.7iT - 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 + 5.24iT - 83T^{2} \) |
| 89 | \( 1 - 14.9T + 89T^{2} \) |
| 97 | \( 1 + 4.44iT - 97T^{2} \) |
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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
−11.02145493194901859885902023156, −10.23970892972287980582582391544, −8.692576938098605556743543492367, −7.78090352218386426196057524651, −7.29162301662007195457792502083, −6.48724781096419677549843793191, −4.90856598343253198185912743718, −3.73970299622832026259108721631, −2.15682926332740490322782216036, −0.41877617072356171753293268452,
2.83598098964990282632676514655, 3.71737393637441797905767167907, 5.16370653955465194142215282014, 5.45324970390636644061466481841, 7.40744180709567331132945366982, 8.191259016259086839951194102281, 9.133209419868067713714478655336, 10.06705981094389970561365768546, 10.82859514215923855997715543253, 11.60450118938691524018470651589