| L(s) = 1 | − 7-s − 4·11-s + 2·13-s + 2·17-s + 4·19-s + 10·29-s − 6·37-s + 6·41-s − 4·43-s + 8·47-s + 49-s + 6·53-s − 4·59-s − 10·61-s + 4·67-s − 16·71-s + 14·73-s + 4·77-s − 8·79-s + 4·83-s − 10·89-s − 2·91-s − 10·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 1.20·11-s + 0.554·13-s + 0.485·17-s + 0.917·19-s + 1.85·29-s − 0.986·37-s + 0.937·41-s − 0.609·43-s + 1.16·47-s + 1/7·49-s + 0.824·53-s − 0.520·59-s − 1.28·61-s + 0.488·67-s − 1.89·71-s + 1.63·73-s + 0.455·77-s − 0.900·79-s + 0.439·83-s − 1.05·89-s − 0.209·91-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.949309537\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.949309537\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−15.52343059547052, −14.93213124777185, −13.98550411251073, −13.88804782830481, −13.29705543841984, −12.56535861145189, −12.25900333266229, −11.62121403515443, −10.92999187363787, −10.35528692483773, −10.07564454317036, −9.336940239880399, −8.671336475978897, −8.213292435969936, −7.493784268583799, −7.107769489978587, −6.232615014409792, −5.763816484526672, −5.106630246478910, −4.526183720000020, −3.638287555091321, −3.011009549980097, −2.489962041156340, −1.403976542785034, −0.5790646714514220,
0.5790646714514220, 1.403976542785034, 2.489962041156340, 3.011009549980097, 3.638287555091321, 4.526183720000020, 5.106630246478910, 5.763816484526672, 6.232615014409792, 7.107769489978587, 7.493784268583799, 8.213292435969936, 8.671336475978897, 9.336940239880399, 10.07564454317036, 10.35528692483773, 10.92999187363787, 11.62121403515443, 12.25900333266229, 12.56535861145189, 13.29705543841984, 13.88804782830481, 13.98550411251073, 14.93213124777185, 15.52343059547052
