| L(s) = 1 | + 5-s − 3·9-s − 4·11-s + 4·13-s − 4·19-s − 3·23-s + 25-s + 29-s − 4·31-s + 7·37-s + 10·41-s − 5·43-s − 3·45-s + 2·47-s − 11·53-s − 4·55-s − 8·61-s + 4·65-s + 4·67-s + 9·71-s − 10·73-s − 79-s + 9·81-s − 16·83-s − 4·89-s − 4·95-s + 8·97-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 9-s − 1.20·11-s + 1.10·13-s − 0.917·19-s − 0.625·23-s + 1/5·25-s + 0.185·29-s − 0.718·31-s + 1.15·37-s + 1.56·41-s − 0.762·43-s − 0.447·45-s + 0.291·47-s − 1.51·53-s − 0.539·55-s − 1.02·61-s + 0.496·65-s + 0.488·67-s + 1.06·71-s − 1.17·73-s − 0.112·79-s + 81-s − 1.75·83-s − 0.423·89-s − 0.410·95-s + 0.812·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 283220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−12.91717952688739, −12.69785064559135, −12.08517363527202, −11.42767675428747, −11.06643035119738, −10.75244417180267, −10.35683870733039, −9.646338759739759, −9.345930095261482, −8.732358515537516, −8.239975945299614, −8.066700438298805, −7.433680365984964, −6.807786873024620, −6.135133773660075, −5.909509880559992, −5.577737066534731, −4.833000059382220, −4.392046734927649, −3.761901666172267, −3.087167294150808, −2.711622775115946, −2.114953786530811, −1.551941858709624, −0.6739932847569450, 0,
0.6739932847569450, 1.551941858709624, 2.114953786530811, 2.711622775115946, 3.087167294150808, 3.761901666172267, 4.392046734927649, 4.833000059382220, 5.577737066534731, 5.909509880559992, 6.135133773660075, 6.807786873024620, 7.433680365984964, 8.066700438298805, 8.239975945299614, 8.732358515537516, 9.345930095261482, 9.646338759739759, 10.35683870733039, 10.75244417180267, 11.06643035119738, 11.42767675428747, 12.08517363527202, 12.69785064559135, 12.91717952688739
