| L(s) = 1 | − 2-s + 1.57·3-s + 4-s − 2.81·5-s − 1.57·6-s + 4.61·7-s − 8-s − 0.531·9-s + 2.81·10-s − 0.167·11-s + 1.57·12-s − 4.61·14-s − 4.42·15-s + 16-s + 2.12·17-s + 0.531·18-s − 19-s − 2.81·20-s + 7.25·21-s + 0.167·22-s + 3.77·23-s − 1.57·24-s + 2.94·25-s − 5.54·27-s + 4.61·28-s + 9.37·29-s + 4.42·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.907·3-s + 0.5·4-s − 1.26·5-s − 0.641·6-s + 1.74·7-s − 0.353·8-s − 0.177·9-s + 0.891·10-s − 0.0504·11-s + 0.453·12-s − 1.23·14-s − 1.14·15-s + 0.250·16-s + 0.515·17-s + 0.125·18-s − 0.229·19-s − 0.630·20-s + 1.58·21-s + 0.0357·22-s + 0.787·23-s − 0.320·24-s + 0.589·25-s − 1.06·27-s + 0.872·28-s + 1.74·29-s + 0.808·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.860629306\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.860629306\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 13 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 1.57T + 3T^{2} \) |
| 5 | \( 1 + 2.81T + 5T^{2} \) |
| 7 | \( 1 - 4.61T + 7T^{2} \) |
| 11 | \( 1 + 0.167T + 11T^{2} \) |
| 17 | \( 1 - 2.12T + 17T^{2} \) |
| 23 | \( 1 - 3.77T + 23T^{2} \) |
| 29 | \( 1 - 9.37T + 29T^{2} \) |
| 31 | \( 1 + 2.76T + 31T^{2} \) |
| 37 | \( 1 + 1.54T + 37T^{2} \) |
| 41 | \( 1 + 0.466T + 41T^{2} \) |
| 43 | \( 1 - 9.52T + 43T^{2} \) |
| 47 | \( 1 + 4.35T + 47T^{2} \) |
| 53 | \( 1 + 6.10T + 53T^{2} \) |
| 59 | \( 1 + 3.90T + 59T^{2} \) |
| 61 | \( 1 - 14.2T + 61T^{2} \) |
| 67 | \( 1 + 1.24T + 67T^{2} \) |
| 71 | \( 1 + 16.1T + 71T^{2} \) |
| 73 | \( 1 - 12.6T + 73T^{2} \) |
| 79 | \( 1 + 9.73T + 79T^{2} \) |
| 83 | \( 1 + 4.41T + 83T^{2} \) |
| 89 | \( 1 - 2.09T + 89T^{2} \) |
| 97 | \( 1 - 16.1T + 97T^{2} \) |
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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
−8.184066072792456417705462720504, −7.57128480706776075451539079781, −7.14066385012461485597256923858, −5.94797221730132927062596331565, −4.98949085979822397397212448805, −4.35324293851806221172382964833, −3.46917589500129819720936488878, −2.70504772733375055161956818585, −1.78216288957183871601661157544, −0.77158857978815563435334094327,
0.77158857978815563435334094327, 1.78216288957183871601661157544, 2.70504772733375055161956818585, 3.46917589500129819720936488878, 4.35324293851806221172382964833, 4.98949085979822397397212448805, 5.94797221730132927062596331565, 7.14066385012461485597256923858, 7.57128480706776075451539079781, 8.184066072792456417705462720504
