| L(s) = 1 | − 0.0716·3-s − 3.69·5-s − 1.67·7-s − 2.99·9-s − 5.28·11-s + 0.264·15-s − 0.533·17-s + 4.65·19-s + 0.119·21-s + 8.51·23-s + 8.65·25-s + 0.429·27-s + 4.48·29-s + 0.899·31-s + 0.378·33-s + 6.18·35-s + 3.17·37-s + 10.2·41-s − 5.24·43-s + 11.0·45-s − 4.09·47-s − 4.19·49-s + 0.0382·51-s − 9.76·53-s + 19.5·55-s − 0.333·57-s − 10.8·59-s + ⋯ |
| L(s) = 1 | − 0.0413·3-s − 1.65·5-s − 0.632·7-s − 0.998·9-s − 1.59·11-s + 0.0683·15-s − 0.129·17-s + 1.06·19-s + 0.0261·21-s + 1.77·23-s + 1.73·25-s + 0.0826·27-s + 0.832·29-s + 0.161·31-s + 0.0659·33-s + 1.04·35-s + 0.521·37-s + 1.59·41-s − 0.799·43-s + 1.64·45-s − 0.597·47-s − 0.599·49-s + 0.00535·51-s − 1.34·53-s + 2.63·55-s − 0.0441·57-s − 1.41·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6413191432\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6413191432\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 0.0716T + 3T^{2} \) |
| 5 | \( 1 + 3.69T + 5T^{2} \) |
| 7 | \( 1 + 1.67T + 7T^{2} \) |
| 11 | \( 1 + 5.28T + 11T^{2} \) |
| 17 | \( 1 + 0.533T + 17T^{2} \) |
| 19 | \( 1 - 4.65T + 19T^{2} \) |
| 23 | \( 1 - 8.51T + 23T^{2} \) |
| 29 | \( 1 - 4.48T + 29T^{2} \) |
| 31 | \( 1 - 0.899T + 31T^{2} \) |
| 37 | \( 1 - 3.17T + 37T^{2} \) |
| 41 | \( 1 - 10.2T + 41T^{2} \) |
| 43 | \( 1 + 5.24T + 43T^{2} \) |
| 47 | \( 1 + 4.09T + 47T^{2} \) |
| 53 | \( 1 + 9.76T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 - 0.994T + 61T^{2} \) |
| 67 | \( 1 - 11.7T + 67T^{2} \) |
| 71 | \( 1 + 12.3T + 71T^{2} \) |
| 73 | \( 1 + 4.70T + 73T^{2} \) |
| 79 | \( 1 - 8.31T + 79T^{2} \) |
| 83 | \( 1 + 12.5T + 83T^{2} \) |
| 89 | \( 1 - 2.12T + 89T^{2} \) |
| 97 | \( 1 - 13.2T + 97T^{2} \) |
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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
−9.526447754800647013089725392468, −8.640588938465754555120005694934, −7.900999344793846507200642275338, −7.41604629357248843332638233150, −6.38736823710482611263613821377, −5.25651293834444904330526263939, −4.58150943246851184797690338278, −3.11872458032245219150800429192, −2.99341698379754426177844989624, −0.55381503067963350397277041325,
0.55381503067963350397277041325, 2.99341698379754426177844989624, 3.11872458032245219150800429192, 4.58150943246851184797690338278, 5.25651293834444904330526263939, 6.38736823710482611263613821377, 7.41604629357248843332638233150, 7.900999344793846507200642275338, 8.640588938465754555120005694934, 9.526447754800647013089725392468
