L(s) = 1 | + 5-s + 4·7-s − 6·13-s + 2·17-s + 4·19-s + 8·23-s + 25-s + 6·29-s + 4·35-s − 6·37-s − 10·41-s − 4·43-s − 8·47-s + 9·49-s − 10·53-s + 6·61-s − 6·65-s − 4·67-s − 14·73-s + 16·79-s − 12·83-s + 2·85-s − 2·89-s − 24·91-s + 4·95-s + 2·97-s + 14·101-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 1.51·7-s − 1.66·13-s + 0.485·17-s + 0.917·19-s + 1.66·23-s + 1/5·25-s + 1.11·29-s + 0.676·35-s − 0.986·37-s − 1.56·41-s − 0.609·43-s − 1.16·47-s + 9/7·49-s − 1.37·53-s + 0.768·61-s − 0.744·65-s − 0.488·67-s − 1.63·73-s + 1.80·79-s − 1.31·83-s + 0.216·85-s − 0.211·89-s − 2.51·91-s + 0.410·95-s + 0.203·97-s + 1.39·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.614650770\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.614650770\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−11.55011761506285336811259028906, −10.48496722029080345949520999093, −9.686400747849369995880150049093, −8.609245026512276947697146238421, −7.67822247307498044046677545125, −6.82423053307536595497404589251, −5.12597312940943561354086695880, −4.92167313563440441781925895939, −2.97865181251010540219662645668, −1.55129913823425505871935815725,
1.55129913823425505871935815725, 2.97865181251010540219662645668, 4.92167313563440441781925895939, 5.12597312940943561354086695880, 6.82423053307536595497404589251, 7.67822247307498044046677545125, 8.609245026512276947697146238421, 9.686400747849369995880150049093, 10.48496722029080345949520999093, 11.55011761506285336811259028906