L(s) = 1 | − 2-s − 4-s + 5-s + 4·7-s + 3·8-s − 10-s + 4·13-s − 4·14-s − 16-s + 6·17-s + 2·19-s − 20-s − 8·23-s + 25-s − 4·26-s − 4·28-s + 2·29-s + 10·31-s − 5·32-s − 6·34-s + 4·35-s + 4·37-s − 2·38-s + 3·40-s + 10·41-s + 8·43-s + 8·46-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.447·5-s + 1.51·7-s + 1.06·8-s − 0.316·10-s + 1.10·13-s − 1.06·14-s − 1/4·16-s + 1.45·17-s + 0.458·19-s − 0.223·20-s − 1.66·23-s + 1/5·25-s − 0.784·26-s − 0.755·28-s + 0.371·29-s + 1.79·31-s − 0.883·32-s − 1.02·34-s + 0.676·35-s + 0.657·37-s − 0.324·38-s + 0.474·40-s + 1.56·41-s + 1.21·43-s + 1.17·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.823877886\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.823877886\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 89 | \( 1 - T \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 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 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−8.235269432272021368145867578328, −8.027321165547185636830477747579, −7.39141308584883466884283376038, −5.99031911402201357680057391006, −5.64540604148670458990049454221, −4.52565269828873131956263229735, −4.13774777206333469982846232579, −2.78196723459830462291795863292, −1.50993434360166562637194045530, −1.02534456621833674318123896945,
1.02534456621833674318123896945, 1.50993434360166562637194045530, 2.78196723459830462291795863292, 4.13774777206333469982846232579, 4.52565269828873131956263229735, 5.64540604148670458990049454221, 5.99031911402201357680057391006, 7.39141308584883466884283376038, 8.027321165547185636830477747579, 8.235269432272021368145867578328