| L(s) = 1 | − 5-s + 11-s − 4·13-s − 6·17-s + 2·19-s + 9·23-s + 25-s − 9·29-s − 4·31-s + 2·37-s − 3·41-s + 11·43-s + 12·47-s − 55-s − 7·61-s + 4·65-s + 5·67-s + 6·71-s + 2·73-s − 16·79-s − 9·83-s + 6·85-s − 9·89-s − 2·95-s − 4·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.301·11-s − 1.10·13-s − 1.45·17-s + 0.458·19-s + 1.87·23-s + 1/5·25-s − 1.67·29-s − 0.718·31-s + 0.328·37-s − 0.468·41-s + 1.67·43-s + 1.75·47-s − 0.134·55-s − 0.896·61-s + 0.496·65-s + 0.610·67-s + 0.712·71-s + 0.234·73-s − 1.80·79-s − 0.987·83-s + 0.650·85-s − 0.953·89-s − 0.205·95-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
| \(L(1)\) |
\(\approx\) |
\(1.288099267\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.288099267\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 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
−13.94148191207230, −13.13287345779594, −12.73609899873038, −12.52860433993932, −11.68404024069724, −11.32389534326829, −10.94794738362240, −10.48446392340252, −9.664412270570643, −9.235155903857455, −8.960150678250078, −8.372844245181251, −7.496374751668251, −7.246688964013767, −6.975422882347614, −6.126125753178434, −5.525758334526917, −5.040153415938034, −4.372439579538273, −4.019006813978509, −3.226921815238823, −2.621753674457707, −2.080381081715793, −1.208221502415283, −0.3752349122712595,
0.3752349122712595, 1.208221502415283, 2.080381081715793, 2.621753674457707, 3.226921815238823, 4.019006813978509, 4.372439579538273, 5.040153415938034, 5.525758334526917, 6.126125753178434, 6.975422882347614, 7.246688964013767, 7.496374751668251, 8.372844245181251, 8.960150678250078, 9.235155903857455, 9.664412270570643, 10.48446392340252, 10.94794738362240, 11.32389534326829, 11.68404024069724, 12.52860433993932, 12.73609899873038, 13.13287345779594, 13.94148191207230
