| L(s) = 1 | − 7-s + 2·11-s + 2·13-s − 6·17-s + 6·23-s + 6·37-s + 2·41-s − 12·43-s − 12·47-s + 49-s + 4·59-s − 10·61-s − 4·67-s + 10·71-s + 14·73-s − 2·77-s + 2·89-s − 2·91-s − 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 6·119-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 0.603·11-s + 0.554·13-s − 1.45·17-s + 1.25·23-s + 0.986·37-s + 0.312·41-s − 1.82·43-s − 1.75·47-s + 1/7·49-s + 0.520·59-s − 1.28·61-s − 0.488·67-s + 1.18·71-s + 1.63·73-s − 0.227·77-s + 0.211·89-s − 0.209·91-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 0.550·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−15.63424599187850, −14.99414374760283, −14.76554605432174, −13.90897489730198, −13.38783500907685, −13.06836040402669, −12.50310179812427, −11.72025384614070, −11.28660321574606, −10.86921369461303, −10.19490634041789, −9.371196455958765, −9.227754242719357, −8.442223635739965, −8.000944665646305, −7.090521943615578, −6.502395410603299, −6.369670995285229, −5.322013161141357, −4.769588462255623, −4.080890446537733, −3.415287998117238, −2.747314044256114, −1.878084067364054, −1.074972092444391, 0,
1.074972092444391, 1.878084067364054, 2.747314044256114, 3.415287998117238, 4.080890446537733, 4.769588462255623, 5.322013161141357, 6.369670995285229, 6.502395410603299, 7.090521943615578, 8.000944665646305, 8.442223635739965, 9.227754242719357, 9.371196455958765, 10.19490634041789, 10.86921369461303, 11.28660321574606, 11.72025384614070, 12.50310179812427, 13.06836040402669, 13.38783500907685, 13.90897489730198, 14.76554605432174, 14.99414374760283, 15.63424599187850
