| L(s) = 1 | + 5-s + 7-s − 3·9-s − 2·11-s − 3·17-s − 6·19-s + 4·23-s − 4·25-s + 7·29-s − 4·31-s + 35-s − 9·37-s + 9·41-s + 10·43-s − 3·45-s + 2·47-s + 49-s − 9·53-s − 2·55-s + 14·59-s + 5·61-s − 3·63-s − 8·67-s − 10·71-s − 7·73-s − 2·77-s − 2·79-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.377·7-s − 9-s − 0.603·11-s − 0.727·17-s − 1.37·19-s + 0.834·23-s − 4/5·25-s + 1.29·29-s − 0.718·31-s + 0.169·35-s − 1.47·37-s + 1.40·41-s + 1.52·43-s − 0.447·45-s + 0.291·47-s + 1/7·49-s − 1.23·53-s − 0.269·55-s + 1.82·59-s + 0.640·61-s − 0.377·63-s − 0.977·67-s − 1.18·71-s − 0.819·73-s − 0.227·77-s − 0.225·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.35007427362570, −13.85184834934540, −13.32029714471681, −12.79078689114654, −12.46961082771634, −11.68604231933434, −11.27812372843930, −10.78497906067836, −10.41019082808438, −9.851102742475955, −9.027548412526805, −8.707952370638324, −8.446391145760418, −7.550865769527653, −7.254148144479013, −6.394541319470813, −5.991116889130471, −5.543406598097633, −4.807481449330410, −4.429674850979427, −3.633853231807175, −2.887597041336190, −2.321921556184432, −1.898414847754667, −0.8203035615236493, 0,
0.8203035615236493, 1.898414847754667, 2.321921556184432, 2.887597041336190, 3.633853231807175, 4.429674850979427, 4.807481449330410, 5.543406598097633, 5.991116889130471, 6.394541319470813, 7.254148144479013, 7.550865769527653, 8.446391145760418, 8.707952370638324, 9.027548412526805, 9.851102742475955, 10.41019082808438, 10.78497906067836, 11.27812372843930, 11.68604231933434, 12.46961082771634, 12.79078689114654, 13.32029714471681, 13.85184834934540, 14.35007427362570
