L(s) = 1 | + 2·3-s + 9-s + 6·11-s − 6·13-s − 6·17-s + 4·19-s + 4·23-s − 4·27-s − 2·29-s + 10·31-s + 12·33-s − 2·37-s − 12·39-s − 2·41-s − 12·43-s − 2·47-s − 7·49-s − 12·51-s − 6·53-s + 8·57-s − 14·59-s + 61-s + 8·67-s + 8·69-s + 10·71-s + 14·73-s − 6·79-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 1/3·9-s + 1.80·11-s − 1.66·13-s − 1.45·17-s + 0.917·19-s + 0.834·23-s − 0.769·27-s − 0.371·29-s + 1.79·31-s + 2.08·33-s − 0.328·37-s − 1.92·39-s − 0.312·41-s − 1.82·43-s − 0.291·47-s − 49-s − 1.68·51-s − 0.824·53-s + 1.05·57-s − 1.82·59-s + 0.128·61-s + 0.977·67-s + 0.963·69-s + 1.18·71-s + 1.63·73-s − 0.675·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 61 | \( 1 - T \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 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
−15.29841821484914, −15.12932034714835, −14.62242535339387, −13.91474720039840, −13.86130545707397, −13.12452126738227, −12.38500716188841, −11.92372524931671, −11.41347649987028, −10.86422669725312, −9.741544218686547, −9.584117435403376, −9.223402815336127, −8.381470785687888, −8.132748422545907, −7.238297589658925, −6.724478967814175, −6.392987636335632, −5.174863047202856, −4.737977224837219, −4.025878504096721, −3.294881184117836, −2.769671788311352, −2.012234518693552, −1.302800211089605, 0,
1.302800211089605, 2.012234518693552, 2.769671788311352, 3.294881184117836, 4.025878504096721, 4.737977224837219, 5.174863047202856, 6.392987636335632, 6.724478967814175, 7.238297589658925, 8.132748422545907, 8.381470785687888, 9.223402815336127, 9.584117435403376, 9.741544218686547, 10.86422669725312, 11.41347649987028, 11.92372524931671, 12.38500716188841, 13.12452126738227, 13.86130545707397, 13.91474720039840, 14.62242535339387, 15.12932034714835, 15.29841821484914