| L(s) = 1 | − 0.623·3-s − 1.00·5-s + 7-s − 2.61·9-s + 0.509·11-s + 0.625·15-s + 1.62·17-s − 7.69·19-s − 0.623·21-s + 1.98·23-s − 3.99·25-s + 3.49·27-s + 1.75·29-s + 10.8·31-s − 0.317·33-s − 1.00·35-s − 7.71·37-s + 8.60·41-s − 12.0·43-s + 2.62·45-s − 9.23·47-s + 49-s − 1.01·51-s − 10.3·53-s − 0.511·55-s + 4.79·57-s + 2.44·59-s + ⋯ |
| L(s) = 1 | − 0.359·3-s − 0.449·5-s + 0.377·7-s − 0.870·9-s + 0.153·11-s + 0.161·15-s + 0.394·17-s − 1.76·19-s − 0.135·21-s + 0.414·23-s − 0.798·25-s + 0.672·27-s + 0.326·29-s + 1.95·31-s − 0.0552·33-s − 0.169·35-s − 1.26·37-s + 1.34·41-s − 1.83·43-s + 0.391·45-s − 1.34·47-s + 0.142·49-s − 0.141·51-s − 1.42·53-s − 0.0689·55-s + 0.635·57-s + 0.317·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.059972981\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.059972981\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 0.623T + 3T^{2} \) |
| 5 | \( 1 + 1.00T + 5T^{2} \) |
| 11 | \( 1 - 0.509T + 11T^{2} \) |
| 17 | \( 1 - 1.62T + 17T^{2} \) |
| 19 | \( 1 + 7.69T + 19T^{2} \) |
| 23 | \( 1 - 1.98T + 23T^{2} \) |
| 29 | \( 1 - 1.75T + 29T^{2} \) |
| 31 | \( 1 - 10.8T + 31T^{2} \) |
| 37 | \( 1 + 7.71T + 37T^{2} \) |
| 41 | \( 1 - 8.60T + 41T^{2} \) |
| 43 | \( 1 + 12.0T + 43T^{2} \) |
| 47 | \( 1 + 9.23T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 - 2.44T + 59T^{2} \) |
| 61 | \( 1 - 13.8T + 61T^{2} \) |
| 67 | \( 1 + 8.97T + 67T^{2} \) |
| 71 | \( 1 - 16.4T + 71T^{2} \) |
| 73 | \( 1 - 10.6T + 73T^{2} \) |
| 79 | \( 1 + 1.34T + 79T^{2} \) |
| 83 | \( 1 + 14.9T + 83T^{2} \) |
| 89 | \( 1 + 6.30T + 89T^{2} \) |
| 97 | \( 1 + 6.70T + 97T^{2} \) |
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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
−7.981771811022364988482726052828, −6.74216205871009303981377903506, −6.49807591104483535716273351327, −5.62527621165737898338596440601, −4.92215631814826997605183257719, −4.30199426935537261928074979397, −3.45297920272527999580022402320, −2.64241259344921697723487470115, −1.70660939620027937517470143843, −0.49443778038251576750943689236,
0.49443778038251576750943689236, 1.70660939620027937517470143843, 2.64241259344921697723487470115, 3.45297920272527999580022402320, 4.30199426935537261928074979397, 4.92215631814826997605183257719, 5.62527621165737898338596440601, 6.49807591104483535716273351327, 6.74216205871009303981377903506, 7.981771811022364988482726052828
