| L(s) = 1 | + 1.51·3-s − 3.62·5-s + 0.892·7-s − 0.710·9-s − 11-s − 2.07·13-s − 5.48·15-s − 1.84·17-s + 0.668·19-s + 1.35·21-s − 3.04·23-s + 8.12·25-s − 5.61·27-s + 6.10·29-s − 0.0259·31-s − 1.51·33-s − 3.23·35-s + 3.85·37-s − 3.13·39-s + 0.603·41-s − 12.0·43-s + 2.57·45-s + 4.57·47-s − 6.20·49-s − 2.79·51-s + 53-s + 3.62·55-s + ⋯ |
| L(s) = 1 | + 0.873·3-s − 1.62·5-s + 0.337·7-s − 0.236·9-s − 0.301·11-s − 0.574·13-s − 1.41·15-s − 0.447·17-s + 0.153·19-s + 0.294·21-s − 0.634·23-s + 1.62·25-s − 1.08·27-s + 1.13·29-s − 0.00466·31-s − 0.263·33-s − 0.546·35-s + 0.634·37-s − 0.502·39-s + 0.0941·41-s − 1.83·43-s + 0.383·45-s + 0.667·47-s − 0.886·49-s − 0.391·51-s + 0.137·53-s + 0.488·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.227032502\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.227032502\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + T \) |
| 53 | \( 1 - T \) |
| good | 3 | \( 1 - 1.51T + 3T^{2} \) |
| 5 | \( 1 + 3.62T + 5T^{2} \) |
| 7 | \( 1 - 0.892T + 7T^{2} \) |
| 13 | \( 1 + 2.07T + 13T^{2} \) |
| 17 | \( 1 + 1.84T + 17T^{2} \) |
| 19 | \( 1 - 0.668T + 19T^{2} \) |
| 23 | \( 1 + 3.04T + 23T^{2} \) |
| 29 | \( 1 - 6.10T + 29T^{2} \) |
| 31 | \( 1 + 0.0259T + 31T^{2} \) |
| 37 | \( 1 - 3.85T + 37T^{2} \) |
| 41 | \( 1 - 0.603T + 41T^{2} \) |
| 43 | \( 1 + 12.0T + 43T^{2} \) |
| 47 | \( 1 - 4.57T + 47T^{2} \) |
| 59 | \( 1 - 3.89T + 59T^{2} \) |
| 61 | \( 1 - 1.00T + 61T^{2} \) |
| 67 | \( 1 - 2.92T + 67T^{2} \) |
| 71 | \( 1 + 11.5T + 71T^{2} \) |
| 73 | \( 1 + 9.62T + 73T^{2} \) |
| 79 | \( 1 - 6.00T + 79T^{2} \) |
| 83 | \( 1 - 3.68T + 83T^{2} \) |
| 89 | \( 1 - 5.66T + 89T^{2} \) |
| 97 | \( 1 + 13.9T + 97T^{2} \) |
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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
−7.84140014275907914610947284574, −7.28680061498923209870413957430, −6.57009568410956655475866591509, −5.54957748566499874142497361947, −4.66381708056625855399396354371, −4.20856236378602582403918861971, −3.33404716279275250376235230090, −2.84063796062123521448128107010, −1.89692414131259318855325744290, −0.48646505915754200865757046333,
0.48646505915754200865757046333, 1.89692414131259318855325744290, 2.84063796062123521448128107010, 3.33404716279275250376235230090, 4.20856236378602582403918861971, 4.66381708056625855399396354371, 5.54957748566499874142497361947, 6.57009568410956655475866591509, 7.28680061498923209870413957430, 7.84140014275907914610947284574
