| L(s) = 1 | − 2-s + 4-s − 5-s − 3·7-s − 8-s + 10-s + 3·11-s + 3·14-s + 16-s + 4·17-s + 7·19-s − 20-s − 3·22-s + 4·23-s + 25-s − 3·28-s + 8·29-s − 10·31-s − 32-s − 4·34-s + 3·35-s + 3·37-s − 7·38-s + 40-s + 2·41-s + 6·43-s + 3·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 1.13·7-s − 0.353·8-s + 0.316·10-s + 0.904·11-s + 0.801·14-s + 1/4·16-s + 0.970·17-s + 1.60·19-s − 0.223·20-s − 0.639·22-s + 0.834·23-s + 1/5·25-s − 0.566·28-s + 1.48·29-s − 1.79·31-s − 0.176·32-s − 0.685·34-s + 0.507·35-s + 0.493·37-s − 1.13·38-s + 0.158·40-s + 0.312·41-s + 0.914·43-s + 0.452·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.462462684\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.462462684\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−16.10690195166084, −15.70672092273217, −15.03433117131660, −14.38667435643190, −13.94846970707849, −13.13622977856842, −12.51237325078212, −12.07819836513114, −11.55651998320826, −10.90921880563384, −10.23009054663542, −9.695433938085428, −9.155190274929328, −8.817085410794386, −7.804692851212129, −7.383043550122666, −6.851606607867194, −6.141603101424147, −5.555310605716259, −4.685393268040891, −3.610946102459242, −3.337679018292014, −2.522260373271639, −1.258899096074474, −0.6753015587107743,
0.6753015587107743, 1.258899096074474, 2.522260373271639, 3.337679018292014, 3.610946102459242, 4.685393268040891, 5.555310605716259, 6.141603101424147, 6.851606607867194, 7.383043550122666, 7.804692851212129, 8.817085410794386, 9.155190274929328, 9.695433938085428, 10.23009054663542, 10.90921880563384, 11.55651998320826, 12.07819836513114, 12.51237325078212, 13.13622977856842, 13.94846970707849, 14.38667435643190, 15.03433117131660, 15.70672092273217, 16.10690195166084
