| L(s) = 1 | − 5-s + 11-s + 4·13-s + 2·17-s + 6·19-s + 6·23-s + 25-s − 4·29-s − 6·31-s + 2·37-s + 6·41-s + 4·43-s + 8·47-s + 6·53-s − 55-s + 12·59-s − 14·61-s − 4·65-s + 4·67-s + 12·73-s + 4·79-s − 2·85-s + 2·89-s − 6·95-s − 4·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.301·11-s + 1.10·13-s + 0.485·17-s + 1.37·19-s + 1.25·23-s + 1/5·25-s − 0.742·29-s − 1.07·31-s + 0.328·37-s + 0.937·41-s + 0.609·43-s + 1.16·47-s + 0.824·53-s − 0.134·55-s + 1.56·59-s − 1.79·61-s − 0.496·65-s + 0.488·67-s + 1.40·73-s + 0.450·79-s − 0.216·85-s + 0.211·89-s − 0.615·95-s − 0.406·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 97020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.365348219\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.365348219\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−13.82468466724492, −13.25114019398713, −12.84924263530129, −12.30543927440513, −11.79403993414708, −11.28379603325924, −10.92125893323740, −10.51154066632573, −9.706088719724374, −9.161432856085702, −9.022007122124445, −8.236275424122164, −7.705619722896286, −7.280758066108104, −6.821965663672538, −6.076715481484526, −5.522705707918152, −5.217360328081958, −4.311266194489695, −3.831628603001683, −3.351161997624943, −2.773907871919949, −1.911990327277122, −1.067487870861373, −0.7070149004286912,
0.7070149004286912, 1.067487870861373, 1.911990327277122, 2.773907871919949, 3.351161997624943, 3.831628603001683, 4.311266194489695, 5.217360328081958, 5.522705707918152, 6.076715481484526, 6.821965663672538, 7.280758066108104, 7.705619722896286, 8.236275424122164, 9.022007122124445, 9.161432856085702, 9.706088719724374, 10.51154066632573, 10.92125893323740, 11.28379603325924, 11.79403993414708, 12.30543927440513, 12.84924263530129, 13.25114019398713, 13.82468466724492
