| L(s) = 1 | − 5-s + 7-s + 2·13-s + 2·17-s + 4·19-s + 25-s − 10·29-s − 35-s + 6·37-s − 6·41-s + 4·43-s + 8·47-s + 49-s − 6·53-s + 4·59-s + 10·61-s − 2·65-s + 4·67-s + 16·71-s + 14·73-s − 8·79-s − 4·83-s − 2·85-s − 10·89-s + 2·91-s − 4·95-s + 10·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s + 0.554·13-s + 0.485·17-s + 0.917·19-s + 1/5·25-s − 1.85·29-s − 0.169·35-s + 0.986·37-s − 0.937·41-s + 0.609·43-s + 1.16·47-s + 1/7·49-s − 0.824·53-s + 0.520·59-s + 1.28·61-s − 0.248·65-s + 0.488·67-s + 1.89·71-s + 1.63·73-s − 0.900·79-s − 0.439·83-s − 0.216·85-s − 1.05·89-s + 0.209·91-s − 0.410·95-s + 1.01·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−12.91626304084082, −12.45066938756519, −11.91389693361757, −11.49024841754307, −11.03930230653752, −10.90277651398510, −10.01085847213404, −9.771219133805598, −9.201160099589314, −8.778927577715295, −8.166233600708190, −7.820415503257223, −7.420720534539275, −6.901695913136545, −6.356825831237832, −5.743041347597301, −5.282579639987317, −4.997971307592247, −4.031710128132975, −3.881763950046987, −3.345154667150756, −2.611585444558892, −2.099814397494389, −1.296101037314817, −0.8810046814437449, 0,
0.8810046814437449, 1.296101037314817, 2.099814397494389, 2.611585444558892, 3.345154667150756, 3.881763950046987, 4.031710128132975, 4.997971307592247, 5.282579639987317, 5.743041347597301, 6.356825831237832, 6.901695913136545, 7.420720534539275, 7.820415503257223, 8.166233600708190, 8.778927577715295, 9.201160099589314, 9.771219133805598, 10.01085847213404, 10.90277651398510, 11.03930230653752, 11.49024841754307, 11.91389693361757, 12.45066938756519, 12.91626304084082
