| L(s) = 1 | + 2·13-s + 2·19-s + 6·29-s + 8·31-s + 4·37-s − 6·41-s + 2·43-s + 6·47-s − 6·53-s + 12·59-s − 8·61-s + 2·67-s − 6·71-s + 2·73-s + 16·79-s − 6·89-s − 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.554·13-s + 0.458·19-s + 1.11·29-s + 1.43·31-s + 0.657·37-s − 0.937·41-s + 0.304·43-s + 0.875·47-s − 0.824·53-s + 1.56·59-s − 1.02·61-s + 0.244·67-s − 0.712·71-s + 0.234·73-s + 1.80·79-s − 0.635·89-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−13.53334923366239, −13.02904860558585, −12.33876302361877, −12.02914540073703, −11.62480664484377, −11.03040928646238, −10.53657696475956, −10.20749601500677, −9.532220788142012, −9.238403565170362, −8.569526065093924, −8.061733901340700, −7.869075551209427, −7.014178753947668, −6.576258682452943, −6.257432581517325, −5.458354351720444, −5.176309782085970, −4.368135234273395, −4.083217557208189, −3.308393572633992, −2.785246417712543, −2.295722731326038, −1.310681234112052, −0.9929506043992309, 0,
0.9929506043992309, 1.310681234112052, 2.295722731326038, 2.785246417712543, 3.308393572633992, 4.083217557208189, 4.368135234273395, 5.176309782085970, 5.458354351720444, 6.257432581517325, 6.576258682452943, 7.014178753947668, 7.869075551209427, 8.061733901340700, 8.569526065093924, 9.238403565170362, 9.532220788142012, 10.20749601500677, 10.53657696475956, 11.03040928646238, 11.62480664484377, 12.02914540073703, 12.33876302361877, 13.02904860558585, 13.53334923366239
