| L(s) = 1 | − 3-s + 9-s − 4·13-s − 4·19-s − 8·23-s − 27-s − 2·29-s − 4·31-s − 2·37-s + 4·39-s + 10·41-s − 43-s + 12·47-s − 7·49-s + 2·53-s + 4·57-s − 8·59-s − 12·61-s + 8·67-s + 8·69-s + 12·71-s + 6·73-s + 81-s + 6·83-s + 2·87-s + 12·89-s + 4·93-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.10·13-s − 0.917·19-s − 1.66·23-s − 0.192·27-s − 0.371·29-s − 0.718·31-s − 0.328·37-s + 0.640·39-s + 1.56·41-s − 0.152·43-s + 1.75·47-s − 49-s + 0.274·53-s + 0.529·57-s − 1.04·59-s − 1.53·61-s + 0.977·67-s + 0.963·69-s + 1.42·71-s + 0.702·73-s + 1/9·81-s + 0.658·83-s + 0.214·87-s + 1.27·89-s + 0.414·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 206400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 206400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−13.10354214906265, −12.65297846890429, −12.23442920465816, −12.12368302236347, −11.23201519382567, −11.05837012930018, −10.41247905068223, −10.12328113414436, −9.405694633418018, −9.258858892628826, −8.515431411119783, −7.810881160293512, −7.673594584756865, −7.032730085611492, −6.457111265709135, −6.010611783493408, −5.594104364185133, −4.937860282862155, −4.517605398229755, −3.939393991663649, −3.499350769787198, −2.503662845744460, −2.204837671936463, −1.554173677774443, −0.6037116282512219, 0,
0.6037116282512219, 1.554173677774443, 2.204837671936463, 2.503662845744460, 3.499350769787198, 3.939393991663649, 4.517605398229755, 4.937860282862155, 5.594104364185133, 6.010611783493408, 6.457111265709135, 7.032730085611492, 7.673594584756865, 7.810881160293512, 8.515431411119783, 9.258858892628826, 9.405694633418018, 10.12328113414436, 10.41247905068223, 11.05837012930018, 11.23201519382567, 12.12368302236347, 12.23442920465816, 12.65297846890429, 13.10354214906265
