L(s) = 1 | + 2-s + 3.08·3-s + 4-s + 0.901·5-s + 3.08·6-s + 5.03·7-s + 8-s + 6.49·9-s + 0.901·10-s + 2.64·11-s + 3.08·12-s − 0.574·13-s + 5.03·14-s + 2.77·15-s + 16-s − 4.29·17-s + 6.49·18-s − 8.71·19-s + 0.901·20-s + 15.5·21-s + 2.64·22-s − 1.58·23-s + 3.08·24-s − 4.18·25-s − 0.574·26-s + 10.7·27-s + 5.03·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.77·3-s + 0.5·4-s + 0.403·5-s + 1.25·6-s + 1.90·7-s + 0.353·8-s + 2.16·9-s + 0.285·10-s + 0.797·11-s + 0.889·12-s − 0.159·13-s + 1.34·14-s + 0.717·15-s + 0.250·16-s − 1.04·17-s + 1.53·18-s − 1.99·19-s + 0.201·20-s + 3.38·21-s + 0.564·22-s − 0.329·23-s + 0.628·24-s − 0.837·25-s − 0.112·26-s + 2.07·27-s + 0.951·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(8.351841369\) |
\(L(\frac12)\) |
\(\approx\) |
\(8.351841369\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| 97 | \( 1 + T \) |
good | 3 | \( 1 - 3.08T + 3T^{2} \) |
| 5 | \( 1 - 0.901T + 5T^{2} \) |
| 7 | \( 1 - 5.03T + 7T^{2} \) |
| 11 | \( 1 - 2.64T + 11T^{2} \) |
| 13 | \( 1 + 0.574T + 13T^{2} \) |
| 17 | \( 1 + 4.29T + 17T^{2} \) |
| 19 | \( 1 + 8.71T + 19T^{2} \) |
| 23 | \( 1 + 1.58T + 23T^{2} \) |
| 29 | \( 1 - 2.71T + 29T^{2} \) |
| 37 | \( 1 + 11.3T + 37T^{2} \) |
| 41 | \( 1 + 0.763T + 41T^{2} \) |
| 43 | \( 1 - 3.77T + 43T^{2} \) |
| 47 | \( 1 - 9.15T + 47T^{2} \) |
| 53 | \( 1 + 5.25T + 53T^{2} \) |
| 59 | \( 1 - 7.17T + 59T^{2} \) |
| 61 | \( 1 - 7.35T + 61T^{2} \) |
| 67 | \( 1 + 8.04T + 67T^{2} \) |
| 71 | \( 1 + 16.5T + 71T^{2} \) |
| 73 | \( 1 - 5.14T + 73T^{2} \) |
| 79 | \( 1 - 1.14T + 79T^{2} \) |
| 83 | \( 1 + 7.58T + 83T^{2} \) |
| 89 | \( 1 + 7.34T + 89T^{2} \) |
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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
−8.226810574136417060667848010863, −7.42403368228739514837603822981, −6.84294897719564451901631263654, −5.91764554612364059888281270803, −4.87175331268752506083163749294, −4.20253913285656800686510292590, −3.91074715331821450349995012363, −2.59290148870905772440526615239, −1.99492907073834624942679365789, −1.59348017672502299411947183155,
1.59348017672502299411947183155, 1.99492907073834624942679365789, 2.59290148870905772440526615239, 3.91074715331821450349995012363, 4.20253913285656800686510292590, 4.87175331268752506083163749294, 5.91764554612364059888281270803, 6.84294897719564451901631263654, 7.42403368228739514837603822981, 8.226810574136417060667848010863