| L(s) = 1 | + 8.64·3-s + 29.5·7-s + 47.7·9-s + 48.2·11-s − 6.18·13-s − 17.9·17-s + 142.·19-s + 255.·21-s + 23·23-s + 179.·27-s + 5.64·29-s + 241.·31-s + 416.·33-s − 136.·37-s − 53.4·39-s − 472.·41-s + 2.10·43-s − 468.·47-s + 530.·49-s − 155.·51-s + 75.5·53-s + 1.23e3·57-s − 589.·59-s − 153.·61-s + 1.41e3·63-s + 707.·67-s + 198.·69-s + ⋯ |
| L(s) = 1 | + 1.66·3-s + 1.59·7-s + 1.76·9-s + 1.32·11-s − 0.131·13-s − 0.256·17-s + 1.71·19-s + 2.65·21-s + 0.208·23-s + 1.27·27-s + 0.0361·29-s + 1.39·31-s + 2.19·33-s − 0.606·37-s − 0.219·39-s − 1.80·41-s + 0.00746·43-s − 1.45·47-s + 1.54·49-s − 0.427·51-s + 0.195·53-s + 2.85·57-s − 1.30·59-s − 0.322·61-s + 2.82·63-s + 1.29·67-s + 0.346·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.547844003\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.547844003\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 - 8.64T + 27T^{2} \) |
| 7 | \( 1 - 29.5T + 343T^{2} \) |
| 11 | \( 1 - 48.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 6.18T + 2.19e3T^{2} \) |
| 17 | \( 1 + 17.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 142.T + 6.85e3T^{2} \) |
| 29 | \( 1 - 5.64T + 2.43e4T^{2} \) |
| 31 | \( 1 - 241.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 136.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 472.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 2.10T + 7.95e4T^{2} \) |
| 47 | \( 1 + 468.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 75.5T + 1.48e5T^{2} \) |
| 59 | \( 1 + 589.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 153.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 707.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 510.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 729.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 80.4T + 4.93e5T^{2} \) |
| 83 | \( 1 - 17.6T + 5.71e5T^{2} \) |
| 89 | \( 1 + 164.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 905.T + 9.12e5T^{2} \) |
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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
−8.431559921285477301868917359917, −8.178720101668731064928225067261, −7.33334796312939545975171309789, −6.64000686161754417742876837170, −5.22460877021993450421277809392, −4.52280557256110487442677281250, −3.64965257674138623562633059211, −2.86472540343032171204499208841, −1.72671620418618516909800172154, −1.25031555224993967134319656357,
1.25031555224993967134319656357, 1.72671620418618516909800172154, 2.86472540343032171204499208841, 3.64965257674138623562633059211, 4.52280557256110487442677281250, 5.22460877021993450421277809392, 6.64000686161754417742876837170, 7.33334796312939545975171309789, 8.178720101668731064928225067261, 8.431559921285477301868917359917
