| L(s) = 1 | − 1.10·3-s − 2.70·5-s + 3.17·7-s − 1.77·9-s + 1.34·11-s − 4.41·13-s + 2.98·15-s − 0.889·17-s + 7.80·19-s − 3.51·21-s − 6.37·23-s + 2.29·25-s + 5.27·27-s + 4.35·29-s + 7.10·31-s − 1.48·33-s − 8.58·35-s + 1.46·37-s + 4.87·39-s − 9.53·41-s + 0.885·43-s + 4.80·45-s + 6.15·47-s + 3.10·49-s + 0.982·51-s − 6.04·53-s − 3.62·55-s + ⋯ |
| L(s) = 1 | − 0.637·3-s − 1.20·5-s + 1.20·7-s − 0.593·9-s + 0.404·11-s − 1.22·13-s + 0.770·15-s − 0.215·17-s + 1.79·19-s − 0.766·21-s − 1.33·23-s + 0.459·25-s + 1.01·27-s + 0.807·29-s + 1.27·31-s − 0.257·33-s − 1.45·35-s + 0.240·37-s + 0.780·39-s − 1.48·41-s + 0.135·43-s + 0.716·45-s + 0.897·47-s + 0.443·49-s + 0.137·51-s − 0.830·53-s − 0.488·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 + 1.10T + 3T^{2} \) |
| 5 | \( 1 + 2.70T + 5T^{2} \) |
| 7 | \( 1 - 3.17T + 7T^{2} \) |
| 11 | \( 1 - 1.34T + 11T^{2} \) |
| 13 | \( 1 + 4.41T + 13T^{2} \) |
| 17 | \( 1 + 0.889T + 17T^{2} \) |
| 19 | \( 1 - 7.80T + 19T^{2} \) |
| 23 | \( 1 + 6.37T + 23T^{2} \) |
| 29 | \( 1 - 4.35T + 29T^{2} \) |
| 31 | \( 1 - 7.10T + 31T^{2} \) |
| 37 | \( 1 - 1.46T + 37T^{2} \) |
| 41 | \( 1 + 9.53T + 41T^{2} \) |
| 43 | \( 1 - 0.885T + 43T^{2} \) |
| 47 | \( 1 - 6.15T + 47T^{2} \) |
| 53 | \( 1 + 6.04T + 53T^{2} \) |
| 59 | \( 1 + 7.23T + 59T^{2} \) |
| 61 | \( 1 - 0.532T + 61T^{2} \) |
| 67 | \( 1 - 11.1T + 67T^{2} \) |
| 71 | \( 1 + 6.84T + 71T^{2} \) |
| 73 | \( 1 + 9.94T + 73T^{2} \) |
| 79 | \( 1 - 8.29T + 79T^{2} \) |
| 83 | \( 1 - 11.7T + 83T^{2} \) |
| 89 | \( 1 + 18.1T + 89T^{2} \) |
| 97 | \( 1 - 3.51T + 97T^{2} \) |
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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
−8.004489576309163206087336840611, −7.51008142958239283300665910474, −6.68258914680079404418031860205, −5.73487957767251057104500080441, −4.91167704756138717620618285690, −4.53286719516424710647601862850, −3.48905486770419409310729970778, −2.51882542446910022742243374783, −1.18479269009430112192421984836, 0,
1.18479269009430112192421984836, 2.51882542446910022742243374783, 3.48905486770419409310729970778, 4.53286719516424710647601862850, 4.91167704756138717620618285690, 5.73487957767251057104500080441, 6.68258914680079404418031860205, 7.51008142958239283300665910474, 8.004489576309163206087336840611
