| L(s) = 1 | − 2-s − 1.64·3-s + 4-s − 0.210·5-s + 1.64·6-s − 4.29·7-s − 8-s − 0.298·9-s + 0.210·10-s − 5.53·11-s − 1.64·12-s + 1.08·13-s + 4.29·14-s + 0.345·15-s + 16-s − 3.80·17-s + 0.298·18-s − 7.09·19-s − 0.210·20-s + 7.06·21-s + 5.53·22-s + 2.95·23-s + 1.64·24-s − 4.95·25-s − 1.08·26-s + 5.42·27-s − 4.29·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.948·3-s + 0.5·4-s − 0.0939·5-s + 0.671·6-s − 1.62·7-s − 0.353·8-s − 0.0994·9-s + 0.0664·10-s − 1.66·11-s − 0.474·12-s + 0.301·13-s + 1.14·14-s + 0.0891·15-s + 0.250·16-s − 0.922·17-s + 0.0703·18-s − 1.62·19-s − 0.0469·20-s + 1.54·21-s + 1.17·22-s + 0.615·23-s + 0.335·24-s − 0.991·25-s − 0.212·26-s + 1.04·27-s − 0.812·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1502 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1502 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1354263193\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1354263193\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 751 | \( 1 - T \) |
| good | 3 | \( 1 + 1.64T + 3T^{2} \) |
| 5 | \( 1 + 0.210T + 5T^{2} \) |
| 7 | \( 1 + 4.29T + 7T^{2} \) |
| 11 | \( 1 + 5.53T + 11T^{2} \) |
| 13 | \( 1 - 1.08T + 13T^{2} \) |
| 17 | \( 1 + 3.80T + 17T^{2} \) |
| 19 | \( 1 + 7.09T + 19T^{2} \) |
| 23 | \( 1 - 2.95T + 23T^{2} \) |
| 29 | \( 1 + 3.08T + 29T^{2} \) |
| 31 | \( 1 - 6.63T + 31T^{2} \) |
| 37 | \( 1 - 4.14T + 37T^{2} \) |
| 41 | \( 1 - 2.73T + 41T^{2} \) |
| 43 | \( 1 + 11.9T + 43T^{2} \) |
| 47 | \( 1 + 6.80T + 47T^{2} \) |
| 53 | \( 1 - 3.05T + 53T^{2} \) |
| 59 | \( 1 + 9.49T + 59T^{2} \) |
| 61 | \( 1 + 2.70T + 61T^{2} \) |
| 67 | \( 1 - 15.7T + 67T^{2} \) |
| 71 | \( 1 + 5.51T + 71T^{2} \) |
| 73 | \( 1 - 3.31T + 73T^{2} \) |
| 79 | \( 1 + 7.06T + 79T^{2} \) |
| 83 | \( 1 - 4.61T + 83T^{2} \) |
| 89 | \( 1 + 11.7T + 89T^{2} \) |
| 97 | \( 1 + 0.942T + 97T^{2} \) |
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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
−9.658527043647859787398337502714, −8.662366278967311244042923632606, −8.005139293565555147641608706438, −6.82617043595428159421078069246, −6.34784722601777330063363417876, −5.65289462688122589807846418837, −4.55900510806417015084120278261, −3.20459604630420638441491926943, −2.32791292429331283065689793419, −0.27788107730216084446024999413,
0.27788107730216084446024999413, 2.32791292429331283065689793419, 3.20459604630420638441491926943, 4.55900510806417015084120278261, 5.65289462688122589807846418837, 6.34784722601777330063363417876, 6.82617043595428159421078069246, 8.005139293565555147641608706438, 8.662366278967311244042923632606, 9.658527043647859787398337502714
