| L(s) = 1 | − 2.92·3-s − 2.56·5-s + 1.38·7-s + 5.57·9-s + 0.939·11-s − 7.18·13-s + 7.52·15-s − 2.97·17-s − 4.99·19-s − 4.05·21-s + 2.14·23-s + 1.60·25-s − 7.55·27-s − 6.66·29-s + 3.09·31-s − 2.75·33-s − 3.55·35-s − 2.80·37-s + 21.0·39-s − 1.51·41-s + 1.62·43-s − 14.3·45-s − 6.67·47-s − 5.08·49-s + 8.70·51-s + 7.26·53-s − 2.41·55-s + ⋯ |
| L(s) = 1 | − 1.69·3-s − 1.14·5-s + 0.523·7-s + 1.85·9-s + 0.283·11-s − 1.99·13-s + 1.94·15-s − 0.720·17-s − 1.14·19-s − 0.885·21-s + 0.447·23-s + 0.320·25-s − 1.45·27-s − 1.23·29-s + 0.555·31-s − 0.479·33-s − 0.601·35-s − 0.460·37-s + 3.37·39-s − 0.236·41-s + 0.247·43-s − 2.13·45-s − 0.973·47-s − 0.725·49-s + 1.21·51-s + 0.997·53-s − 0.325·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.05723203172\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05723203172\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.92T + 3T^{2} \) |
| 5 | \( 1 + 2.56T + 5T^{2} \) |
| 7 | \( 1 - 1.38T + 7T^{2} \) |
| 11 | \( 1 - 0.939T + 11T^{2} \) |
| 13 | \( 1 + 7.18T + 13T^{2} \) |
| 17 | \( 1 + 2.97T + 17T^{2} \) |
| 19 | \( 1 + 4.99T + 19T^{2} \) |
| 23 | \( 1 - 2.14T + 23T^{2} \) |
| 29 | \( 1 + 6.66T + 29T^{2} \) |
| 31 | \( 1 - 3.09T + 31T^{2} \) |
| 37 | \( 1 + 2.80T + 37T^{2} \) |
| 41 | \( 1 + 1.51T + 41T^{2} \) |
| 43 | \( 1 - 1.62T + 43T^{2} \) |
| 47 | \( 1 + 6.67T + 47T^{2} \) |
| 53 | \( 1 - 7.26T + 53T^{2} \) |
| 59 | \( 1 + 1.63T + 59T^{2} \) |
| 61 | \( 1 + 7.36T + 61T^{2} \) |
| 67 | \( 1 - 7.38T + 67T^{2} \) |
| 71 | \( 1 - 0.0410T + 71T^{2} \) |
| 73 | \( 1 - 3.58T + 73T^{2} \) |
| 79 | \( 1 + 4.57T + 79T^{2} \) |
| 83 | \( 1 + 16.4T + 83T^{2} \) |
| 89 | \( 1 + 16.2T + 89T^{2} \) |
| 97 | \( 1 + 3.86T + 97T^{2} \) |
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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
−7.64689288409703997975761614722, −6.98315730305290177770171825154, −6.61901196639532381541737535083, −5.61102425823211238890027956556, −4.97605035596799198682530353246, −4.47562687782318937025673069724, −3.92130326941124506345914616713, −2.57692346593814113577730330400, −1.52717681791413607482377479344, −0.13015023446220623707631397900,
0.13015023446220623707631397900, 1.52717681791413607482377479344, 2.57692346593814113577730330400, 3.92130326941124506345914616713, 4.47562687782318937025673069724, 4.97605035596799198682530353246, 5.61102425823211238890027956556, 6.61901196639532381541737535083, 6.98315730305290177770171825154, 7.64689288409703997975761614722
