| L(s) = 1 | + 11-s + 4·13-s − 4·17-s + 19-s + 5·23-s − 3·29-s + 5·31-s − 6·37-s + 2·41-s − 4·43-s − 7·49-s − 9·53-s − 11·61-s + 67-s + 2·71-s − 3·73-s − 17·79-s + 3·83-s − 7·89-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.301·11-s + 1.10·13-s − 0.970·17-s + 0.229·19-s + 1.04·23-s − 0.557·29-s + 0.898·31-s − 0.986·37-s + 0.312·41-s − 0.609·43-s − 49-s − 1.23·53-s − 1.40·61-s + 0.122·67-s + 0.237·71-s − 0.351·73-s − 1.91·79-s + 0.329·83-s − 0.741·89-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.152632276\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.152632276\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 + 17 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 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
−14.17162773885629, −13.54901875914087, −13.25508561311105, −12.75214595759005, −12.16636498639506, −11.50703753425541, −11.19730723913987, −10.73453816117311, −10.16366226464869, −9.492611019857938, −9.046949060228429, −8.576998162162655, −8.108318907345801, −7.400818609381240, −6.831368687382276, −6.380116772540504, −5.875408025600201, −5.151822366199615, −4.581661142279805, −4.068087171416614, −3.243764886779278, −2.954506330842734, −1.865658048727571, −1.429236952189277, −0.4964548776642145,
0.4964548776642145, 1.429236952189277, 1.865658048727571, 2.954506330842734, 3.243764886779278, 4.068087171416614, 4.581661142279805, 5.151822366199615, 5.875408025600201, 6.380116772540504, 6.831368687382276, 7.400818609381240, 8.108318907345801, 8.576998162162655, 9.046949060228429, 9.492611019857938, 10.16366226464869, 10.73453816117311, 11.19730723913987, 11.50703753425541, 12.16636498639506, 12.75214595759005, 13.25508561311105, 13.54901875914087, 14.17162773885629
