| L(s) = 1 | − 3.10·3-s − 3.31·5-s + 6.61·9-s − 0.792·11-s + 6.10·13-s + 10.2·15-s + 0.423·17-s + 0.177·19-s + 5.55·23-s + 5.98·25-s − 11.2·27-s + 6.87·29-s + 1.03·31-s + 2.45·33-s + 10.9·37-s − 18.9·39-s − 41-s + 7.60·43-s − 21.9·45-s − 0.719·47-s − 1.31·51-s + 9.70·53-s + 2.62·55-s − 0.551·57-s + 8.31·59-s + 9.24·61-s − 20.2·65-s + ⋯ |
| L(s) = 1 | − 1.79·3-s − 1.48·5-s + 2.20·9-s − 0.238·11-s + 1.69·13-s + 2.65·15-s + 0.102·17-s + 0.0407·19-s + 1.15·23-s + 1.19·25-s − 2.15·27-s + 1.27·29-s + 0.186·31-s + 0.427·33-s + 1.79·37-s − 3.02·39-s − 0.156·41-s + 1.15·43-s − 3.26·45-s − 0.104·47-s − 0.183·51-s + 1.33·53-s + 0.353·55-s − 0.0730·57-s + 1.08·59-s + 1.18·61-s − 2.50·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9429738925\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9429738925\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 + 3.10T + 3T^{2} \) |
| 5 | \( 1 + 3.31T + 5T^{2} \) |
| 11 | \( 1 + 0.792T + 11T^{2} \) |
| 13 | \( 1 - 6.10T + 13T^{2} \) |
| 17 | \( 1 - 0.423T + 17T^{2} \) |
| 19 | \( 1 - 0.177T + 19T^{2} \) |
| 23 | \( 1 - 5.55T + 23T^{2} \) |
| 29 | \( 1 - 6.87T + 29T^{2} \) |
| 31 | \( 1 - 1.03T + 31T^{2} \) |
| 37 | \( 1 - 10.9T + 37T^{2} \) |
| 43 | \( 1 - 7.60T + 43T^{2} \) |
| 47 | \( 1 + 0.719T + 47T^{2} \) |
| 53 | \( 1 - 9.70T + 53T^{2} \) |
| 59 | \( 1 - 8.31T + 59T^{2} \) |
| 61 | \( 1 - 9.24T + 61T^{2} \) |
| 67 | \( 1 + 14.6T + 67T^{2} \) |
| 71 | \( 1 - 6.50T + 71T^{2} \) |
| 73 | \( 1 - 15.7T + 73T^{2} \) |
| 79 | \( 1 - 0.0105T + 79T^{2} \) |
| 83 | \( 1 - 7.04T + 83T^{2} \) |
| 89 | \( 1 + 16.3T + 89T^{2} \) |
| 97 | \( 1 + 12.9T + 97T^{2} \) |
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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
−7.77744425984653796420283679843, −6.89880707938082066533827084511, −6.53210262434017508034476586063, −5.71160432520846383040008160811, −5.11605388657757858964479463875, −4.22003750036715995102745283201, −3.94084384783357204518374548954, −2.82015063396702658128137698433, −1.10473739950142081428321693002, −0.68365212194153441821987358470,
0.68365212194153441821987358470, 1.10473739950142081428321693002, 2.82015063396702658128137698433, 3.94084384783357204518374548954, 4.22003750036715995102745283201, 5.11605388657757858964479463875, 5.71160432520846383040008160811, 6.53210262434017508034476586063, 6.89880707938082066533827084511, 7.77744425984653796420283679843
