L(s) = 1 | + 3-s + 5-s + 3.64·7-s + 9-s + 5.64·11-s − 1.64·13-s + 15-s + 19-s + 3.64·21-s − 2·23-s + 25-s + 27-s + 4.35·29-s + 2·31-s + 5.64·33-s + 3.64·35-s + 8.93·37-s − 1.64·39-s − 3.64·41-s − 10.9·43-s + 45-s − 6·47-s + 6.29·49-s + 4·53-s + 5.64·55-s + 57-s + 7.29·59-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1.37·7-s + 0.333·9-s + 1.70·11-s − 0.456·13-s + 0.258·15-s + 0.229·19-s + 0.795·21-s − 0.417·23-s + 0.200·25-s + 0.192·27-s + 0.808·29-s + 0.359·31-s + 0.982·33-s + 0.616·35-s + 1.46·37-s − 0.263·39-s − 0.569·41-s − 1.66·43-s + 0.149·45-s − 0.875·47-s + 0.898·49-s + 0.549·53-s + 0.761·55-s + 0.132·57-s + 0.949·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.658989202\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.658989202\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 - T \) |
good | 7 | \( 1 - 3.64T + 7T^{2} \) |
| 11 | \( 1 - 5.64T + 11T^{2} \) |
| 13 | \( 1 + 1.64T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 23 | \( 1 + 2T + 23T^{2} \) |
| 29 | \( 1 - 4.35T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 8.93T + 37T^{2} \) |
| 41 | \( 1 + 3.64T + 41T^{2} \) |
| 43 | \( 1 + 10.9T + 43T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 - 4T + 53T^{2} \) |
| 59 | \( 1 - 7.29T + 59T^{2} \) |
| 61 | \( 1 - 0.708T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 7.29T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 15.8T + 83T^{2} \) |
| 89 | \( 1 - 14.2T + 89T^{2} \) |
| 97 | \( 1 + 8.22T + 97T^{2} \) |
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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
−8.378309426815539897520853714466, −7.74741352423048493119381581866, −6.86459974175772536417860130436, −6.28668354697129778483231024563, −5.24226187610104879422397118322, −4.55045502353599921608921653401, −3.88111592719966943130413809287, −2.80265042955694524879887531705, −1.81118068223647028460996115851, −1.18900520423033290457766561742,
1.18900520423033290457766561742, 1.81118068223647028460996115851, 2.80265042955694524879887531705, 3.88111592719966943130413809287, 4.55045502353599921608921653401, 5.24226187610104879422397118322, 6.28668354697129778483231024563, 6.86459974175772536417860130436, 7.74741352423048493119381581866, 8.378309426815539897520853714466