| L(s) = 1 | + 3.23·3-s + 1.87·5-s + 2.94·7-s + 7.45·9-s + 3.98·11-s − 13-s + 6.04·15-s − 5.02·17-s − 0.0553·19-s + 9.53·21-s − 1.07·23-s − 1.50·25-s + 14.4·27-s + 29-s − 10.2·31-s + 12.8·33-s + 5.51·35-s − 9.39·37-s − 3.23·39-s + 7.68·41-s + 1.72·43-s + 13.9·45-s + 12.7·47-s + 1.69·49-s − 16.2·51-s + 7.71·53-s + 7.44·55-s + ⋯ |
| L(s) = 1 | + 1.86·3-s + 0.836·5-s + 1.11·7-s + 2.48·9-s + 1.20·11-s − 0.277·13-s + 1.56·15-s − 1.21·17-s − 0.0127·19-s + 2.08·21-s − 0.223·23-s − 0.300·25-s + 2.77·27-s + 0.185·29-s − 1.83·31-s + 2.24·33-s + 0.932·35-s − 1.54·37-s − 0.517·39-s + 1.20·41-s + 0.262·43-s + 2.07·45-s + 1.85·47-s + 0.241·49-s − 2.27·51-s + 1.06·53-s + 1.00·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.940268474\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.940268474\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 - 3.23T + 3T^{2} \) |
| 5 | \( 1 - 1.87T + 5T^{2} \) |
| 7 | \( 1 - 2.94T + 7T^{2} \) |
| 11 | \( 1 - 3.98T + 11T^{2} \) |
| 17 | \( 1 + 5.02T + 17T^{2} \) |
| 19 | \( 1 + 0.0553T + 19T^{2} \) |
| 23 | \( 1 + 1.07T + 23T^{2} \) |
| 31 | \( 1 + 10.2T + 31T^{2} \) |
| 37 | \( 1 + 9.39T + 37T^{2} \) |
| 41 | \( 1 - 7.68T + 41T^{2} \) |
| 43 | \( 1 - 1.72T + 43T^{2} \) |
| 47 | \( 1 - 12.7T + 47T^{2} \) |
| 53 | \( 1 - 7.71T + 53T^{2} \) |
| 59 | \( 1 + 2.19T + 59T^{2} \) |
| 61 | \( 1 + 9.35T + 61T^{2} \) |
| 67 | \( 1 - 8.67T + 67T^{2} \) |
| 71 | \( 1 - 13.1T + 71T^{2} \) |
| 73 | \( 1 - 4.26T + 73T^{2} \) |
| 79 | \( 1 + 5.03T + 79T^{2} \) |
| 83 | \( 1 + 10.2T + 83T^{2} \) |
| 89 | \( 1 + 2.62T + 89T^{2} \) |
| 97 | \( 1 + 5.93T + 97T^{2} \) |
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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
−8.290823450718935467115225494405, −7.35823943037850512886944575274, −7.02985291552764533559856280766, −5.99999911101588028835107043439, −5.03671621792313985765981212920, −4.13309295276241846153783599436, −3.75075872318246310393907212717, −2.50046634973550700113409601446, −2.00627000024304855397284778389, −1.37665781114158077325794606502,
1.37665781114158077325794606502, 2.00627000024304855397284778389, 2.50046634973550700113409601446, 3.75075872318246310393907212717, 4.13309295276241846153783599436, 5.03671621792313985765981212920, 5.99999911101588028835107043439, 7.02985291552764533559856280766, 7.35823943037850512886944575274, 8.290823450718935467115225494405
