| L(s) = 1 | + 2·3-s + 9-s − 4·11-s + 6·13-s + 4·17-s − 6·19-s + 4·23-s − 4·27-s + 6·29-s − 4·31-s − 8·33-s − 6·37-s + 12·39-s + 4·41-s + 12·43-s + 12·47-s + 8·51-s + 6·53-s − 12·57-s − 6·59-s + 6·61-s + 12·67-s + 8·69-s + 8·71-s − 11·81-s + 6·83-s + 12·87-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/3·9-s − 1.20·11-s + 1.66·13-s + 0.970·17-s − 1.37·19-s + 0.834·23-s − 0.769·27-s + 1.11·29-s − 0.718·31-s − 1.39·33-s − 0.986·37-s + 1.92·39-s + 0.624·41-s + 1.82·43-s + 1.75·47-s + 1.12·51-s + 0.824·53-s − 1.58·57-s − 0.781·59-s + 0.768·61-s + 1.46·67-s + 0.963·69-s + 0.949·71-s − 1.22·81-s + 0.658·83-s + 1.28·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.080512782\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.080512782\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 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
−14.02083534651560, −13.62749488070000, −13.07790294261108, −12.64810495135405, −12.28712003922897, −11.37937731161237, −10.79361040405125, −10.65451345414877, −10.04120496400392, −9.226832146841538, −8.929569261480467, −8.397793289106824, −8.084033776878765, −7.530006665757367, −6.944381185189216, −6.247864362290445, −5.638949650935150, −5.277614971730908, −4.314520804129739, −3.825452569002301, −3.347628955986751, −2.563813902314228, −2.341548606470388, −1.349019589747580, −0.6311815850262817,
0.6311815850262817, 1.349019589747580, 2.341548606470388, 2.563813902314228, 3.347628955986751, 3.825452569002301, 4.314520804129739, 5.277614971730908, 5.638949650935150, 6.247864362290445, 6.944381185189216, 7.530006665757367, 8.084033776878765, 8.397793289106824, 8.929569261480467, 9.226832146841538, 10.04120496400392, 10.65451345414877, 10.79361040405125, 11.37937731161237, 12.28712003922897, 12.64810495135405, 13.07790294261108, 13.62749488070000, 14.02083534651560
