L(s) = 1 | + 2-s − 4-s − 3·8-s − 3·9-s + 2·13-s − 16-s + 6·17-s − 3·18-s + 4·19-s − 4·23-s + 2·26-s − 6·29-s − 8·31-s + 5·32-s + 6·34-s + 3·36-s + 2·37-s + 4·38-s − 2·41-s + 4·43-s − 4·46-s + 12·47-s − 7·49-s − 2·52-s + 2·53-s − 6·58-s + 4·59-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.06·8-s − 9-s + 0.554·13-s − 1/4·16-s + 1.45·17-s − 0.707·18-s + 0.917·19-s − 0.834·23-s + 0.392·26-s − 1.11·29-s − 1.43·31-s + 0.883·32-s + 1.02·34-s + 1/2·36-s + 0.328·37-s + 0.648·38-s − 0.312·41-s + 0.609·43-s − 0.589·46-s + 1.75·47-s − 49-s − 0.277·52-s + 0.274·53-s − 0.787·58-s + 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.856771715\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.856771715\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 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 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 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
−8.700136523246462655071585370550, −8.032357569215232401617509130184, −7.25345373525116928335761342083, −6.01863453952841743401384036248, −5.62267982183930582090539793665, −5.04438306355119198762685190347, −3.64867610629817493009082817690, −3.57615991727962736404141628215, −2.30183692911339057669317387952, −0.74278936026352163297831552325,
0.74278936026352163297831552325, 2.30183692911339057669317387952, 3.57615991727962736404141628215, 3.64867610629817493009082817690, 5.04438306355119198762685190347, 5.62267982183930582090539793665, 6.01863453952841743401384036248, 7.25345373525116928335761342083, 8.032357569215232401617509130184, 8.700136523246462655071585370550