| L(s) = 1 | − 7-s + 6·11-s − 13-s + 3·17-s + 4·19-s − 3·23-s + 3·29-s + 5·31-s − 10·37-s − 9·41-s − 43-s + 49-s − 9·53-s + 9·59-s − 11·61-s − 4·67-s + 12·71-s + 10·73-s − 6·77-s − 10·79-s − 9·83-s + 6·89-s + 91-s − 14·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 1.80·11-s − 0.277·13-s + 0.727·17-s + 0.917·19-s − 0.625·23-s + 0.557·29-s + 0.898·31-s − 1.64·37-s − 1.40·41-s − 0.152·43-s + 1/7·49-s − 1.23·53-s + 1.17·59-s − 1.40·61-s − 0.488·67-s + 1.42·71-s + 1.17·73-s − 0.683·77-s − 1.12·79-s − 0.987·83-s + 0.635·89-s + 0.104·91-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.459876351\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.459876351\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| good | 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−13.78716787381406, −13.53037418191423, −12.46378012356031, −12.25294479447790, −11.94011490984350, −11.43582483171468, −10.81340378813932, −10.15049670225095, −9.754728154479399, −9.421763107094895, −8.813131406142780, −8.256133844540092, −7.830185514800177, −6.970198366141581, −6.726890992853891, −6.304580358853607, −5.468141501102483, −5.172471876196062, −4.277859923213728, −3.902231427952444, −3.224665335631101, −2.836054593057288, −1.693058405085098, −1.407723486416719, −0.5058801693400264,
0.5058801693400264, 1.407723486416719, 1.693058405085098, 2.836054593057288, 3.224665335631101, 3.902231427952444, 4.277859923213728, 5.172471876196062, 5.468141501102483, 6.304580358853607, 6.726890992853891, 6.970198366141581, 7.830185514800177, 8.256133844540092, 8.813131406142780, 9.421763107094895, 9.754728154479399, 10.15049670225095, 10.81340378813932, 11.43582483171468, 11.94011490984350, 12.25294479447790, 12.46378012356031, 13.53037418191423, 13.78716787381406
