L(s) = 1 | + 2·3-s + 4·5-s + 9-s + 8·15-s + 2·17-s − 2·19-s − 8·23-s + 11·25-s − 4·27-s + 2·29-s + 4·31-s − 6·37-s + 2·41-s − 8·43-s + 4·45-s − 4·47-s + 4·51-s − 10·53-s − 4·57-s + 6·59-s − 4·61-s + 12·67-s − 16·69-s + 14·73-s + 22·75-s + 8·79-s − 11·81-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 1.78·5-s + 1/3·9-s + 2.06·15-s + 0.485·17-s − 0.458·19-s − 1.66·23-s + 11/5·25-s − 0.769·27-s + 0.371·29-s + 0.718·31-s − 0.986·37-s + 0.312·41-s − 1.21·43-s + 0.596·45-s − 0.583·47-s + 0.560·51-s − 1.37·53-s − 0.529·57-s + 0.781·59-s − 0.512·61-s + 1.46·67-s − 1.92·69-s + 1.63·73-s + 2.54·75-s + 0.900·79-s − 1.22·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.866540363\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.866540363\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−9.835763078329112662938551691511, −9.676028605922657643630885041741, −8.589598276995096865608741769842, −8.037894346653340821155380595819, −6.69299348503088401254902880103, −5.96146004102990052123544748972, −4.99083605129769608898331014735, −3.56791835941750047835108384805, −2.48337724162238959661867349800, −1.73163501201493931998851829334,
1.73163501201493931998851829334, 2.48337724162238959661867349800, 3.56791835941750047835108384805, 4.99083605129769608898331014735, 5.96146004102990052123544748972, 6.69299348503088401254902880103, 8.037894346653340821155380595819, 8.589598276995096865608741769842, 9.676028605922657643630885041741, 9.835763078329112662938551691511