| L(s) = 1 | − i·3-s + (1 + 2i)5-s + 4i·7-s − 9-s + 4·11-s + (2 − i)15-s − 4i·17-s + 4·21-s − 4i·23-s + (−3 + 4i)25-s + i·27-s + 6·29-s − 4·31-s − 4i·33-s + (−8 + 4i)35-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + (0.447 + 0.894i)5-s + 1.51i·7-s − 0.333·9-s + 1.20·11-s + (0.516 − 0.258i)15-s − 0.970i·17-s + 0.872·21-s − 0.834i·23-s + (−0.600 + 0.800i)25-s + 0.192i·27-s + 1.11·29-s − 0.718·31-s − 0.696i·33-s + (−1.35 + 0.676i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.30995 + 0.309238i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.30995 + 0.309238i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (-1 - 2i)T \) |
| good | 7 | \( 1 - 4iT - 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 4iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 8iT - 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 + 12iT - 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 8iT - 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 + 8iT - 97T^{2} \) |
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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
−11.86527860078859643386918401116, −11.69557978991296139162751627427, −10.23854141178758539465137344087, −9.206828541264297597886211202999, −8.403970479013721844609992079615, −6.90294503198921896812726799886, −6.32533802292600679739143347223, −5.16314623769505609217332084251, −3.17641775447271335113341091605, −2.03750830747473422943185169035,
1.32498126435075574265836892451, 3.74930876962917927796877532822, 4.48294582052482520316776606537, 5.83634324969267790664286206408, 7.02478871152944469914671863621, 8.294472639280201139352829703901, 9.277170527550531457196157822216, 10.08901862853590514828989988385, 10.94791882407485644600954965544, 12.05595988028134139540206773746
