| L(s) = 1 | + 5-s + 5·11-s − 3·17-s + 5·19-s − 6·23-s + 25-s + 10·29-s − 2·31-s + 4·37-s + 3·41-s + 3·43-s − 4·47-s − 7·49-s + 6·53-s + 5·55-s + 3·59-s + 2·61-s − 11·67-s + 14·71-s − 15·73-s + 10·79-s + 12·83-s − 3·85-s − 14·89-s + 5·95-s − 13·97-s + 12·101-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.50·11-s − 0.727·17-s + 1.14·19-s − 1.25·23-s + 1/5·25-s + 1.85·29-s − 0.359·31-s + 0.657·37-s + 0.468·41-s + 0.457·43-s − 0.583·47-s − 49-s + 0.824·53-s + 0.674·55-s + 0.390·59-s + 0.256·61-s − 1.34·67-s + 1.66·71-s − 1.75·73-s + 1.12·79-s + 1.31·83-s − 0.325·85-s − 1.48·89-s + 0.512·95-s − 1.31·97-s + 1.19·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.294962796\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.294962796\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 + 15 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−8.721986862372531466472811439682, −7.978200054339079524025434372864, −7.01473875379533385520716772330, −6.40472777915298073926030895749, −5.77374847167872842672541376014, −4.71844238354751501924695230175, −4.02581030722532199561603893552, −3.04958657678263978554504940068, −1.97229666763931659198908570606, −0.959453154224763848219262354022,
0.959453154224763848219262354022, 1.97229666763931659198908570606, 3.04958657678263978554504940068, 4.02581030722532199561603893552, 4.71844238354751501924695230175, 5.77374847167872842672541376014, 6.40472777915298073926030895749, 7.01473875379533385520716772330, 7.978200054339079524025434372864, 8.721986862372531466472811439682
