L(s) = 1 | − 2·7-s + 2·11-s + 2·17-s + 4·19-s + 2·29-s + 8·31-s + 4·37-s − 8·41-s − 8·43-s + 8·47-s − 3·49-s − 10·53-s − 6·59-s + 2·61-s − 12·67-s + 12·71-s + 2·73-s − 4·77-s + 8·79-s + 4·83-s − 12·89-s − 10·97-s + 10·101-s − 10·103-s + 12·107-s + 10·109-s + 6·113-s + ⋯ |
L(s) = 1 | − 0.755·7-s + 0.603·11-s + 0.485·17-s + 0.917·19-s + 0.371·29-s + 1.43·31-s + 0.657·37-s − 1.24·41-s − 1.21·43-s + 1.16·47-s − 3/7·49-s − 1.37·53-s − 0.781·59-s + 0.256·61-s − 1.46·67-s + 1.42·71-s + 0.234·73-s − 0.455·77-s + 0.900·79-s + 0.439·83-s − 1.27·89-s − 1.01·97-s + 0.995·101-s − 0.985·103-s + 1.16·107-s + 0.957·109-s + 0.564·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.893360635\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.893360635\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 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
−7.977022162170601162717385016934, −7.12942120472983786257929764756, −6.51890325540431486867551333265, −5.96228168395430582623933126484, −5.07276593123802042471929932552, −4.36386726865745771885203372220, −3.37010468324439478155641422955, −2.96272925490350914639778649878, −1.71809051730295497621980546526, −0.71333718150537513818455008431,
0.71333718150537513818455008431, 1.71809051730295497621980546526, 2.96272925490350914639778649878, 3.37010468324439478155641422955, 4.36386726865745771885203372220, 5.07276593123802042471929932552, 5.96228168395430582623933126484, 6.51890325540431486867551333265, 7.12942120472983786257929764756, 7.977022162170601162717385016934