| L(s) = 1 | − 5-s + 11-s + 6·13-s + 5·17-s − 7·19-s − 5·23-s + 25-s − 3·29-s + 6·31-s − 5·37-s + 6·41-s + 7·43-s − 11·47-s − 8·53-s − 55-s − 9·59-s + 2·61-s − 6·65-s + 8·67-s + 5·71-s + 16·73-s + 8·79-s − 6·83-s − 5·85-s + 14·89-s + 7·95-s − 3·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.301·11-s + 1.66·13-s + 1.21·17-s − 1.60·19-s − 1.04·23-s + 1/5·25-s − 0.557·29-s + 1.07·31-s − 0.821·37-s + 0.937·41-s + 1.06·43-s − 1.60·47-s − 1.09·53-s − 0.134·55-s − 1.17·59-s + 0.256·61-s − 0.744·65-s + 0.977·67-s + 0.593·71-s + 1.87·73-s + 0.900·79-s − 0.658·83-s − 0.542·85-s + 1.48·89-s + 0.718·95-s − 0.304·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.884462847\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.884462847\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + 11 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
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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
−12.49730969575065, −12.04504368367913, −11.50147588415243, −11.06705032831000, −10.78276402724918, −10.25001062406114, −9.795048140311414, −9.221666851323988, −8.812637115110744, −8.255510901985766, −7.931158621110109, −7.692414197167012, −6.718798430541900, −6.480727657943843, −6.051972171853593, −5.609674392083666, −4.906875882432527, −4.350811164946939, −3.946026946682015, −3.417930086098912, −3.122207886461193, −2.088744310393792, −1.823255255816410, −0.9788338505124994, −0.5075473177035604,
0.5075473177035604, 0.9788338505124994, 1.823255255816410, 2.088744310393792, 3.122207886461193, 3.417930086098912, 3.946026946682015, 4.350811164946939, 4.906875882432527, 5.609674392083666, 6.051972171853593, 6.480727657943843, 6.718798430541900, 7.692414197167012, 7.931158621110109, 8.255510901985766, 8.812637115110744, 9.221666851323988, 9.795048140311414, 10.25001062406114, 10.78276402724918, 11.06705032831000, 11.50147588415243, 12.04504368367913, 12.49730969575065
