| L(s) = 1 | − 3-s + 0.846·7-s + 9-s + 5.97·11-s + 0.955·13-s + 3.74·17-s + 1.01·19-s − 0.846·21-s + 4.41·23-s − 27-s − 2.34·29-s + 31-s − 5.97·33-s + 9.76·37-s − 0.955·39-s − 7.12·41-s + 7.96·43-s + 11.8·47-s − 6.28·49-s − 3.74·51-s + 1.93·53-s − 1.01·57-s + 5.22·59-s + 13.4·61-s + 0.846·63-s + 13.3·67-s − 4.41·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.319·7-s + 0.333·9-s + 1.80·11-s + 0.265·13-s + 0.908·17-s + 0.232·19-s − 0.184·21-s + 0.919·23-s − 0.192·27-s − 0.436·29-s + 0.179·31-s − 1.04·33-s + 1.60·37-s − 0.153·39-s − 1.11·41-s + 1.21·43-s + 1.72·47-s − 0.897·49-s − 0.524·51-s + 0.266·53-s − 0.134·57-s + 0.680·59-s + 1.71·61-s + 0.106·63-s + 1.62·67-s − 0.530·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.427871015\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.427871015\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 - 0.846T + 7T^{2} \) |
| 11 | \( 1 - 5.97T + 11T^{2} \) |
| 13 | \( 1 - 0.955T + 13T^{2} \) |
| 17 | \( 1 - 3.74T + 17T^{2} \) |
| 19 | \( 1 - 1.01T + 19T^{2} \) |
| 23 | \( 1 - 4.41T + 23T^{2} \) |
| 29 | \( 1 + 2.34T + 29T^{2} \) |
| 37 | \( 1 - 9.76T + 37T^{2} \) |
| 41 | \( 1 + 7.12T + 41T^{2} \) |
| 43 | \( 1 - 7.96T + 43T^{2} \) |
| 47 | \( 1 - 11.8T + 47T^{2} \) |
| 53 | \( 1 - 1.93T + 53T^{2} \) |
| 59 | \( 1 - 5.22T + 59T^{2} \) |
| 61 | \( 1 - 13.4T + 61T^{2} \) |
| 67 | \( 1 - 13.3T + 67T^{2} \) |
| 71 | \( 1 + 7.87T + 71T^{2} \) |
| 73 | \( 1 + 12.9T + 73T^{2} \) |
| 79 | \( 1 + 14.0T + 79T^{2} \) |
| 83 | \( 1 + 17.0T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 - 12.1T + 97T^{2} \) |
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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
−7.47821766942804072367372927237, −7.07418305606584285101698169634, −6.25544162417371869595486058147, −5.76125680543001302294853936816, −4.99238183943170201101937336225, −4.14730218035278473458213970040, −3.67255487764583328493204503182, −2.60970015542919939678004772170, −1.40460589781959841753961897568, −0.898002964662371027807389617559,
0.898002964662371027807389617559, 1.40460589781959841753961897568, 2.60970015542919939678004772170, 3.67255487764583328493204503182, 4.14730218035278473458213970040, 4.99238183943170201101937336225, 5.76125680543001302294853936816, 6.25544162417371869595486058147, 7.07418305606584285101698169634, 7.47821766942804072367372927237
