L(s) = 1 | + 3-s + 9-s − 6·11-s − 13-s − 2·17-s − 4·23-s + 27-s + 6·29-s − 4·31-s − 6·33-s − 2·37-s − 39-s + 4·43-s − 10·47-s − 7·49-s − 2·51-s − 10·53-s + 6·59-s + 6·61-s − 12·67-s − 4·69-s + 2·71-s − 6·73-s − 16·79-s + 81-s + 6·83-s + 6·87-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1/3·9-s − 1.80·11-s − 0.277·13-s − 0.485·17-s − 0.834·23-s + 0.192·27-s + 1.11·29-s − 0.718·31-s − 1.04·33-s − 0.328·37-s − 0.160·39-s + 0.609·43-s − 1.45·47-s − 49-s − 0.280·51-s − 1.37·53-s + 0.781·59-s + 0.768·61-s − 1.46·67-s − 0.481·69-s + 0.237·71-s − 0.702·73-s − 1.80·79-s + 1/9·81-s + 0.658·83-s + 0.643·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.050252026\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.050252026\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 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
−14.27265306913580, −13.77347396127033, −13.22119520888971, −12.79518455634419, −12.51964819943766, −11.65526070498697, −11.27056632733451, −10.50546678447645, −10.24689530708022, −9.736783537463042, −9.128057086546804, −8.425910465904788, −8.146645441546150, −7.593838860479773, −7.114448879153043, −6.400597624142150, −5.832413135721906, −5.094902401000551, −4.734249433550188, −4.056829696261974, −3.217272059065566, −2.801617290266937, −2.162178471009372, −1.527021438885852, −0.3147546326783569,
0.3147546326783569, 1.527021438885852, 2.162178471009372, 2.801617290266937, 3.217272059065566, 4.056829696261974, 4.734249433550188, 5.094902401000551, 5.832413135721906, 6.400597624142150, 7.114448879153043, 7.593838860479773, 8.146645441546150, 8.425910465904788, 9.128057086546804, 9.736783537463042, 10.24689530708022, 10.50546678447645, 11.27056632733451, 11.65526070498697, 12.51964819943766, 12.79518455634419, 13.22119520888971, 13.77347396127033, 14.27265306913580