| L(s) = 1 | − 2.27·2-s + 3.16·4-s + 1.72·5-s − 2.65·8-s − 3.91·10-s + 2.87·11-s + 1.56·13-s − 0.308·16-s + 5.43·17-s − 19-s + 5.45·20-s − 6.52·22-s + 6.09·23-s − 2.03·25-s − 3.55·26-s − 3.17·29-s + 1.03·31-s + 6.00·32-s − 12.3·34-s + 8.44·37-s + 2.27·38-s − 4.56·40-s + 10.4·41-s − 3.21·43-s + 9.09·44-s − 13.8·46-s − 5.50·47-s + ⋯ |
| L(s) = 1 | − 1.60·2-s + 1.58·4-s + 0.770·5-s − 0.937·8-s − 1.23·10-s + 0.865·11-s + 0.434·13-s − 0.0770·16-s + 1.31·17-s − 0.229·19-s + 1.21·20-s − 1.39·22-s + 1.27·23-s − 0.406·25-s − 0.697·26-s − 0.589·29-s + 0.185·31-s + 1.06·32-s − 2.11·34-s + 1.38·37-s + 0.368·38-s − 0.721·40-s + 1.62·41-s − 0.491·43-s + 1.37·44-s − 2.04·46-s − 0.803·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8379 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8379 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.351824757\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.351824757\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 2.27T + 2T^{2} \) |
| 5 | \( 1 - 1.72T + 5T^{2} \) |
| 11 | \( 1 - 2.87T + 11T^{2} \) |
| 13 | \( 1 - 1.56T + 13T^{2} \) |
| 17 | \( 1 - 5.43T + 17T^{2} \) |
| 23 | \( 1 - 6.09T + 23T^{2} \) |
| 29 | \( 1 + 3.17T + 29T^{2} \) |
| 31 | \( 1 - 1.03T + 31T^{2} \) |
| 37 | \( 1 - 8.44T + 37T^{2} \) |
| 41 | \( 1 - 10.4T + 41T^{2} \) |
| 43 | \( 1 + 3.21T + 43T^{2} \) |
| 47 | \( 1 + 5.50T + 47T^{2} \) |
| 53 | \( 1 - 12.9T + 53T^{2} \) |
| 59 | \( 1 + 6.21T + 59T^{2} \) |
| 61 | \( 1 + 9.76T + 61T^{2} \) |
| 67 | \( 1 + 2.55T + 67T^{2} \) |
| 71 | \( 1 + 12.3T + 71T^{2} \) |
| 73 | \( 1 - 10.5T + 73T^{2} \) |
| 79 | \( 1 - 12.9T + 79T^{2} \) |
| 83 | \( 1 - 5.38T + 83T^{2} \) |
| 89 | \( 1 - 7.45T + 89T^{2} \) |
| 97 | \( 1 - 2.45T + 97T^{2} \) |
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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
−7.73575679926657310700198533123, −7.47942619499719180570783328701, −6.43592776891434409768082064997, −6.12023335881199160314518021926, −5.21033507514426782138601590008, −4.19235400317573440501035783232, −3.21192769630445904974836759766, −2.29216600714286218094399615477, −1.42030586738914079603968062221, −0.820073071840093394200506315724,
0.820073071840093394200506315724, 1.42030586738914079603968062221, 2.29216600714286218094399615477, 3.21192769630445904974836759766, 4.19235400317573440501035783232, 5.21033507514426782138601590008, 6.12023335881199160314518021926, 6.43592776891434409768082064997, 7.47942619499719180570783328701, 7.73575679926657310700198533123
