| L(s) = 1 | + 5-s − 5·7-s − 3·9-s + 3·11-s − 4·13-s + 3·17-s + 23-s − 4·25-s − 4·29-s + 4·31-s − 5·35-s − 4·37-s + 6·41-s − 9·43-s − 3·45-s + 7·47-s + 18·49-s + 3·55-s − 2·59-s − 11·61-s + 15·63-s − 4·65-s + 14·67-s − 7·73-s − 15·77-s + 8·79-s + 9·81-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.88·7-s − 9-s + 0.904·11-s − 1.10·13-s + 0.727·17-s + 0.208·23-s − 4/5·25-s − 0.742·29-s + 0.718·31-s − 0.845·35-s − 0.657·37-s + 0.937·41-s − 1.37·43-s − 0.447·45-s + 1.02·47-s + 18/7·49-s + 0.404·55-s − 0.260·59-s − 1.40·61-s + 1.88·63-s − 0.496·65-s + 1.71·67-s − 0.819·73-s − 1.70·77-s + 0.900·79-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 66424 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66424 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−14.37105036491096, −13.86448854567663, −13.58011432422372, −12.88794589473252, −12.39183878264597, −12.06487798527860, −11.57825365756669, −10.85778008965999, −10.23502939905180, −9.677961796690359, −9.554973145494873, −8.990517138456151, −8.407340168714202, −7.644759740570986, −7.072656452594334, −6.627651298243952, −5.952559499320450, −5.773834582373666, −5.052570578956489, −4.196356866185249, −3.559740213012401, −3.075869812676293, −2.554261103280187, −1.805489000742306, −0.7228728662782117, 0,
0.7228728662782117, 1.805489000742306, 2.554261103280187, 3.075869812676293, 3.559740213012401, 4.196356866185249, 5.052570578956489, 5.773834582373666, 5.952559499320450, 6.627651298243952, 7.072656452594334, 7.644759740570986, 8.407340168714202, 8.990517138456151, 9.554973145494873, 9.677961796690359, 10.23502939905180, 10.85778008965999, 11.57825365756669, 12.06487798527860, 12.39183878264597, 12.88794589473252, 13.58011432422372, 13.86448854567663, 14.37105036491096
