L(s) = 1 | − 3-s + 5-s − 1.71·7-s + 9-s − 5.77·11-s − 4.04·13-s − 15-s + 2.26·17-s − 5.06·19-s + 1.71·21-s − 8.09·23-s + 25-s − 27-s + 6.95·29-s − 1.98·31-s + 5.77·33-s − 1.71·35-s + 10.0·37-s + 4.04·39-s − 1.61·41-s + 2.42·43-s + 45-s + 5.66·47-s − 4.04·49-s − 2.26·51-s + 0.372·53-s − 5.77·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.649·7-s + 0.333·9-s − 1.74·11-s − 1.12·13-s − 0.258·15-s + 0.550·17-s − 1.16·19-s + 0.374·21-s − 1.68·23-s + 0.200·25-s − 0.192·27-s + 1.29·29-s − 0.356·31-s + 1.00·33-s − 0.290·35-s + 1.66·37-s + 0.647·39-s − 0.251·41-s + 0.370·43-s + 0.149·45-s + 0.826·47-s − 0.578·49-s − 0.317·51-s + 0.0511·53-s − 0.778·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8215624316\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8215624316\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 67 | \( 1 - T \) |
good | 7 | \( 1 + 1.71T + 7T^{2} \) |
| 11 | \( 1 + 5.77T + 11T^{2} \) |
| 13 | \( 1 + 4.04T + 13T^{2} \) |
| 17 | \( 1 - 2.26T + 17T^{2} \) |
| 19 | \( 1 + 5.06T + 19T^{2} \) |
| 23 | \( 1 + 8.09T + 23T^{2} \) |
| 29 | \( 1 - 6.95T + 29T^{2} \) |
| 31 | \( 1 + 1.98T + 31T^{2} \) |
| 37 | \( 1 - 10.0T + 37T^{2} \) |
| 41 | \( 1 + 1.61T + 41T^{2} \) |
| 43 | \( 1 - 2.42T + 43T^{2} \) |
| 47 | \( 1 - 5.66T + 47T^{2} \) |
| 53 | \( 1 - 0.372T + 53T^{2} \) |
| 59 | \( 1 - 6.04T + 59T^{2} \) |
| 61 | \( 1 - 7.06T + 61T^{2} \) |
| 71 | \( 1 - 0.432T + 71T^{2} \) |
| 73 | \( 1 - 12.0T + 73T^{2} \) |
| 79 | \( 1 - 4.63T + 79T^{2} \) |
| 83 | \( 1 - 15.5T + 83T^{2} \) |
| 89 | \( 1 - 8.07T + 89T^{2} \) |
| 97 | \( 1 + 6.07T + 97T^{2} \) |
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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
−8.170301507613267230362602201642, −7.84364967895136309608070755603, −6.84488710915663227461676891396, −6.18581772812907107258655415667, −5.49426218212708051729517172345, −4.84959172977185171326595720775, −3.95818498799722231655901672320, −2.68994241593238459248311044013, −2.19953885546602347175958089228, −0.49998839772493631048311501410,
0.49998839772493631048311501410, 2.19953885546602347175958089228, 2.68994241593238459248311044013, 3.95818498799722231655901672320, 4.84959172977185171326595720775, 5.49426218212708051729517172345, 6.18581772812907107258655415667, 6.84488710915663227461676891396, 7.84364967895136309608070755603, 8.170301507613267230362602201642