| L(s) = 1 | + 5-s − 2·7-s + 4·13-s + 4·19-s − 4·23-s + 25-s + 6·29-s − 8·31-s − 2·35-s − 10·37-s − 6·41-s − 6·43-s − 12·47-s − 3·49-s + 10·53-s − 4·59-s + 14·61-s + 4·65-s − 8·67-s + 8·71-s + 4·73-s − 14·83-s + 18·89-s − 8·91-s + 4·95-s − 10·97-s + 101-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.755·7-s + 1.10·13-s + 0.917·19-s − 0.834·23-s + 1/5·25-s + 1.11·29-s − 1.43·31-s − 0.338·35-s − 1.64·37-s − 0.937·41-s − 0.914·43-s − 1.75·47-s − 3/7·49-s + 1.37·53-s − 0.520·59-s + 1.79·61-s + 0.496·65-s − 0.977·67-s + 0.949·71-s + 0.468·73-s − 1.53·83-s + 1.90·89-s − 0.838·91-s + 0.410·95-s − 1.01·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.930947808\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.930947808\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 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
−14.58549290071075, −14.06024400469008, −13.66479285956495, −13.14403617868056, −12.77531815411598, −12.03055343727191, −11.64547238362996, −11.05503841577863, −10.26874147027955, −10.09755347995220, −9.497776024930888, −8.798635110214733, −8.479412236560593, −7.801573861198537, −6.992473605276546, −6.620506751391633, −6.096951125935730, −5.381900397072431, −5.035487769022197, −4.025517905214228, −3.421087256762020, −3.105137161329405, −2.011915039366615, −1.507923306335824, −0.4976231618608407,
0.4976231618608407, 1.507923306335824, 2.011915039366615, 3.105137161329405, 3.421087256762020, 4.025517905214228, 5.035487769022197, 5.381900397072431, 6.096951125935730, 6.620506751391633, 6.992473605276546, 7.801573861198537, 8.479412236560593, 8.798635110214733, 9.497776024930888, 10.09755347995220, 10.26874147027955, 11.05503841577863, 11.64547238362996, 12.03055343727191, 12.77531815411598, 13.14403617868056, 13.66479285956495, 14.06024400469008, 14.58549290071075
