| L(s) = 1 | − 3.12·3-s − 3.70·5-s − 1.25·7-s + 6.79·9-s + 5.85·11-s − 5.70·13-s + 11.5·15-s − 0.899·17-s + 2.13·19-s + 3.93·21-s − 1.38·23-s + 8.74·25-s − 11.8·27-s − 1.64·29-s + 3.15·31-s − 18.3·33-s + 4.66·35-s − 9.50·37-s + 17.8·39-s − 7.18·41-s − 10.1·43-s − 25.1·45-s − 0.589·47-s − 5.41·49-s + 2.81·51-s + 3.68·53-s − 21.7·55-s + ⋯ |
| L(s) = 1 | − 1.80·3-s − 1.65·5-s − 0.475·7-s + 2.26·9-s + 1.76·11-s − 1.58·13-s + 2.99·15-s − 0.218·17-s + 0.489·19-s + 0.859·21-s − 0.289·23-s + 1.74·25-s − 2.28·27-s − 0.305·29-s + 0.565·31-s − 3.19·33-s + 0.788·35-s − 1.56·37-s + 2.85·39-s − 1.12·41-s − 1.54·43-s − 3.75·45-s − 0.0859·47-s − 0.773·49-s + 0.394·51-s + 0.505·53-s − 2.92·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2546203267\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2546203267\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 - T \) |
| good | 3 | \( 1 + 3.12T + 3T^{2} \) |
| 5 | \( 1 + 3.70T + 5T^{2} \) |
| 7 | \( 1 + 1.25T + 7T^{2} \) |
| 11 | \( 1 - 5.85T + 11T^{2} \) |
| 13 | \( 1 + 5.70T + 13T^{2} \) |
| 17 | \( 1 + 0.899T + 17T^{2} \) |
| 19 | \( 1 - 2.13T + 19T^{2} \) |
| 23 | \( 1 + 1.38T + 23T^{2} \) |
| 29 | \( 1 + 1.64T + 29T^{2} \) |
| 31 | \( 1 - 3.15T + 31T^{2} \) |
| 37 | \( 1 + 9.50T + 37T^{2} \) |
| 41 | \( 1 + 7.18T + 41T^{2} \) |
| 43 | \( 1 + 10.1T + 43T^{2} \) |
| 47 | \( 1 + 0.589T + 47T^{2} \) |
| 53 | \( 1 - 3.68T + 53T^{2} \) |
| 59 | \( 1 + 0.265T + 59T^{2} \) |
| 61 | \( 1 + 12.0T + 61T^{2} \) |
| 67 | \( 1 - 0.794T + 67T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 - 8.75T + 73T^{2} \) |
| 79 | \( 1 + 10.6T + 79T^{2} \) |
| 83 | \( 1 - 1.59T + 83T^{2} \) |
| 89 | \( 1 - 0.0712T + 89T^{2} \) |
| 97 | \( 1 - 16.8T + 97T^{2} \) |
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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
−8.298397483118059462633893161719, −7.40445905019741545846704649487, −6.77541310558656376789549570008, −6.55236611770631922476085486756, −5.31733633066541217520048630004, −4.73849617353776018349781706345, −4.03970904321888326069300730731, −3.32758795219557217046469525900, −1.53406495195441014773309231739, −0.32887477180386019200652978396,
0.32887477180386019200652978396, 1.53406495195441014773309231739, 3.32758795219557217046469525900, 4.03970904321888326069300730731, 4.73849617353776018349781706345, 5.31733633066541217520048630004, 6.55236611770631922476085486756, 6.77541310558656376789549570008, 7.40445905019741545846704649487, 8.298397483118059462633893161719
