L(s) = 1 | − 5-s − 7-s + 4·13-s + 4·19-s − 6·23-s + 25-s − 6·29-s − 4·31-s + 35-s + 2·37-s − 6·41-s + 4·43-s − 6·47-s + 49-s + 6·53-s + 10·61-s − 4·65-s + 2·67-s − 12·71-s − 8·73-s + 16·79-s − 12·83-s + 18·89-s − 4·91-s − 4·95-s − 10·97-s + 101-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 0.377·7-s + 1.10·13-s + 0.917·19-s − 1.25·23-s + 1/5·25-s − 1.11·29-s − 0.718·31-s + 0.169·35-s + 0.328·37-s − 0.937·41-s + 0.609·43-s − 0.875·47-s + 1/7·49-s + 0.824·53-s + 1.28·61-s − 0.496·65-s + 0.244·67-s − 1.42·71-s − 0.936·73-s + 1.80·79-s − 1.31·83-s + 1.90·89-s − 0.419·91-s − 0.410·95-s − 1.01·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.421784606\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.421784606\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 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
−13.36878987968143, −12.92034725284302, −12.31980666118908, −11.88975932062858, −11.36363900302464, −11.11643783390089, −10.41514445740683, −9.986806930364055, −9.510410869162302, −8.935109334835236, −8.518889636003574, −7.980152547998483, −7.458115459676188, −7.073884372016659, −6.360714338327783, −5.947678465757307, −5.449051644681048, −4.881119826728470, −4.047850674781551, −3.742849123685859, −3.306514520561297, −2.546865154745990, −1.824056527099039, −1.204758143444436, −0.3681523748190980,
0.3681523748190980, 1.204758143444436, 1.824056527099039, 2.546865154745990, 3.306514520561297, 3.742849123685859, 4.047850674781551, 4.881119826728470, 5.449051644681048, 5.947678465757307, 6.360714338327783, 7.073884372016659, 7.458115459676188, 7.980152547998483, 8.518889636003574, 8.935109334835236, 9.510410869162302, 9.986806930364055, 10.41514445740683, 11.11643783390089, 11.36363900302464, 11.88975932062858, 12.31980666118908, 12.92034725284302, 13.36878987968143