| L(s) = 1 | − 2·5-s + 7-s + 6·11-s − 7·13-s + 2·17-s − 25-s − 10·29-s − 8·31-s − 2·35-s − 7·37-s + 9·43-s − 8·47-s − 6·49-s + 2·53-s − 12·55-s + 4·59-s − 7·61-s + 14·65-s + 3·67-s − 12·71-s − 14·73-s + 6·77-s − 8·79-s − 14·83-s − 4·85-s + 12·89-s − 7·91-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.377·7-s + 1.80·11-s − 1.94·13-s + 0.485·17-s − 1/5·25-s − 1.85·29-s − 1.43·31-s − 0.338·35-s − 1.15·37-s + 1.37·43-s − 1.16·47-s − 6/7·49-s + 0.274·53-s − 1.61·55-s + 0.520·59-s − 0.896·61-s + 1.73·65-s + 0.366·67-s − 1.42·71-s − 1.63·73-s + 0.683·77-s − 0.900·79-s − 1.53·83-s − 0.433·85-s + 1.27·89-s − 0.733·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38088 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9397346313\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9397346313\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 7 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−14.77107552229728, −14.34900989546999, −14.22412314868611, −13.07144270600823, −12.72333045152295, −12.07455246513829, −11.66967882657179, −11.45869929641373, −10.70964324202852, −9.986651249915029, −9.467838932455808, −9.055880628919192, −8.464502165496380, −7.559842303710356, −7.390047710008156, −6.981062706124351, −6.024018156230879, −5.511748447632728, −4.729826232686507, −4.248710887634296, −3.640940305821422, −3.110157016154654, −1.979006624551724, −1.573149478594243, −0.3496720125355538,
0.3496720125355538, 1.573149478594243, 1.979006624551724, 3.110157016154654, 3.640940305821422, 4.248710887634296, 4.729826232686507, 5.511748447632728, 6.024018156230879, 6.981062706124351, 7.390047710008156, 7.559842303710356, 8.464502165496380, 9.055880628919192, 9.467838932455808, 9.986651249915029, 10.70964324202852, 11.45869929641373, 11.66967882657179, 12.07455246513829, 12.72333045152295, 13.07144270600823, 14.22412314868611, 14.34900989546999, 14.77107552229728
