| L(s) = 1 | − 2.44·3-s − 3·5-s + 0.449·7-s + 2.99·9-s + 2.44·11-s + 5.89·13-s + 7.34·15-s − 4.89·17-s − 3.55·19-s − 1.10·21-s + 4·25-s + 7.89·29-s + 10.8·31-s − 5.99·33-s − 1.34·35-s + 8·37-s − 14.4·39-s + 7.89·41-s − 6.89·43-s − 8.99·45-s − 1.55·47-s − 6.79·49-s + 11.9·51-s + 8.79·53-s − 7.34·55-s + 8.69·57-s − 5.34·59-s + ⋯ |
| L(s) = 1 | − 1.41·3-s − 1.34·5-s + 0.169·7-s + 0.999·9-s + 0.738·11-s + 1.63·13-s + 1.89·15-s − 1.18·17-s − 0.814·19-s − 0.240·21-s + 0.800·25-s + 1.46·29-s + 1.95·31-s − 1.04·33-s − 0.227·35-s + 1.31·37-s − 2.31·39-s + 1.23·41-s − 1.05·43-s − 1.34·45-s − 0.226·47-s − 0.971·49-s + 1.68·51-s + 1.20·53-s − 0.990·55-s + 1.15·57-s − 0.696·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8931884369\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8931884369\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + 2.44T + 3T^{2} \) |
| 5 | \( 1 + 3T + 5T^{2} \) |
| 7 | \( 1 - 0.449T + 7T^{2} \) |
| 11 | \( 1 - 2.44T + 11T^{2} \) |
| 13 | \( 1 - 5.89T + 13T^{2} \) |
| 17 | \( 1 + 4.89T + 17T^{2} \) |
| 19 | \( 1 + 3.55T + 19T^{2} \) |
| 29 | \( 1 - 7.89T + 29T^{2} \) |
| 31 | \( 1 - 10.8T + 31T^{2} \) |
| 37 | \( 1 - 8T + 37T^{2} \) |
| 41 | \( 1 - 7.89T + 41T^{2} \) |
| 43 | \( 1 + 6.89T + 43T^{2} \) |
| 47 | \( 1 + 1.55T + 47T^{2} \) |
| 53 | \( 1 - 8.79T + 53T^{2} \) |
| 59 | \( 1 + 5.34T + 59T^{2} \) |
| 61 | \( 1 - 5.89T + 61T^{2} \) |
| 67 | \( 1 + 13.7T + 67T^{2} \) |
| 71 | \( 1 - 4.44T + 71T^{2} \) |
| 73 | \( 1 + 1.89T + 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 + 6.89T + 83T^{2} \) |
| 89 | \( 1 - 8.79T + 89T^{2} \) |
| 97 | \( 1 + 1.89T + 97T^{2} \) |
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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
−7.78098523552231746436921124402, −6.83541994547435757093443458090, −6.29680431745563089588601182998, −6.05573371932215340017507843255, −4.69976944413607947119600354203, −4.46131312179153461526264297037, −3.80485494160127383364013876299, −2.75243464786618541414837244980, −1.29578320510573281359593748324, −0.57114632521050694568744861385,
0.57114632521050694568744861385, 1.29578320510573281359593748324, 2.75243464786618541414837244980, 3.80485494160127383364013876299, 4.46131312179153461526264297037, 4.69976944413607947119600354203, 6.05573371932215340017507843255, 6.29680431745563089588601182998, 6.83541994547435757093443458090, 7.78098523552231746436921124402
