| L(s) = 1 | − 7-s − 4·11-s − 2·13-s − 6·17-s − 6·29-s − 6·37-s + 10·41-s + 12·47-s + 49-s + 6·53-s + 4·59-s − 2·61-s − 8·67-s − 4·71-s − 10·73-s + 4·77-s + 12·83-s + 10·89-s + 2·91-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 6·119-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 1.20·11-s − 0.554·13-s − 1.45·17-s − 1.11·29-s − 0.986·37-s + 1.56·41-s + 1.75·47-s + 1/7·49-s + 0.824·53-s + 0.520·59-s − 0.256·61-s − 0.977·67-s − 0.474·71-s − 1.17·73-s + 0.455·77-s + 1.31·83-s + 1.05·89-s + 0.209·91-s + 1.42·97-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.85135878557489, −13.42385373432304, −12.99140195809831, −12.73837038187292, −11.91973615278503, −11.71279361588196, −10.76983306833739, −10.67814807455106, −10.20546846493091, −9.424172832974061, −9.047683071396936, −8.679225164177102, −7.858441377771180, −7.441374511328177, −7.098061720806715, −6.357532583663192, −5.837184748357358, −5.319708322547202, −4.750852710393199, −4.165626498590165, −3.615383497270133, −2.737919347904118, −2.418307111930656, −1.794992337476674, −0.6667432600571413, 0,
0.6667432600571413, 1.794992337476674, 2.418307111930656, 2.737919347904118, 3.615383497270133, 4.165626498590165, 4.750852710393199, 5.319708322547202, 5.837184748357358, 6.357532583663192, 7.098061720806715, 7.441374511328177, 7.858441377771180, 8.679225164177102, 9.047683071396936, 9.424172832974061, 10.20546846493091, 10.67814807455106, 10.76983306833739, 11.71279361588196, 11.91973615278503, 12.73837038187292, 12.99140195809831, 13.42385373432304, 13.85135878557489
