| L(s) = 1 | + 1.61·2-s − 1.86·3-s + 0.607·4-s − 3.01·6-s + 2.29·7-s − 2.24·8-s + 0.486·9-s + 3.61·11-s − 1.13·12-s − 3.56·13-s + 3.70·14-s − 4.84·16-s + 5.22·17-s + 0.786·18-s − 6.37·19-s − 4.28·21-s + 5.83·22-s + 8.92·23-s + 4.19·24-s − 5.75·26-s + 4.69·27-s + 1.39·28-s − 0.531·29-s + 5.07·31-s − 3.32·32-s − 6.74·33-s + 8.43·34-s + ⋯ |
| L(s) = 1 | + 1.14·2-s − 1.07·3-s + 0.303·4-s − 1.23·6-s + 0.868·7-s − 0.795·8-s + 0.162·9-s + 1.08·11-s − 0.327·12-s − 0.989·13-s + 0.991·14-s − 1.21·16-s + 1.26·17-s + 0.185·18-s − 1.46·19-s − 0.935·21-s + 1.24·22-s + 1.86·23-s + 0.857·24-s − 1.12·26-s + 0.903·27-s + 0.263·28-s − 0.0986·29-s + 0.911·31-s − 0.587·32-s − 1.17·33-s + 1.44·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.049114299\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.049114299\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 71 | \( 1 + T \) |
| good | 2 | \( 1 - 1.61T + 2T^{2} \) |
| 3 | \( 1 + 1.86T + 3T^{2} \) |
| 7 | \( 1 - 2.29T + 7T^{2} \) |
| 11 | \( 1 - 3.61T + 11T^{2} \) |
| 13 | \( 1 + 3.56T + 13T^{2} \) |
| 17 | \( 1 - 5.22T + 17T^{2} \) |
| 19 | \( 1 + 6.37T + 19T^{2} \) |
| 23 | \( 1 - 8.92T + 23T^{2} \) |
| 29 | \( 1 + 0.531T + 29T^{2} \) |
| 31 | \( 1 - 5.07T + 31T^{2} \) |
| 37 | \( 1 + 3.41T + 37T^{2} \) |
| 41 | \( 1 - 7.62T + 41T^{2} \) |
| 43 | \( 1 + 0.250T + 43T^{2} \) |
| 47 | \( 1 - 6.37T + 47T^{2} \) |
| 53 | \( 1 - 8.02T + 53T^{2} \) |
| 59 | \( 1 - 13.2T + 59T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 + 11.2T + 67T^{2} \) |
| 73 | \( 1 - 16.5T + 73T^{2} \) |
| 79 | \( 1 - 5.50T + 79T^{2} \) |
| 83 | \( 1 - 15.5T + 83T^{2} \) |
| 89 | \( 1 + 2.21T + 89T^{2} \) |
| 97 | \( 1 - 7.19T + 97T^{2} \) |
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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
−9.235241798379834158610295134089, −8.583213525392232027826537768055, −7.41247013570278693973997554829, −6.54324192561393372528670455523, −5.89793469906341366259208510796, −5.01732760600499938648857060594, −4.68529578936512507093124664890, −3.66528280400216712596090263322, −2.48339331408512725582067192394, −0.894798139239909221663278138700,
0.894798139239909221663278138700, 2.48339331408512725582067192394, 3.66528280400216712596090263322, 4.68529578936512507093124664890, 5.01732760600499938648857060594, 5.89793469906341366259208510796, 6.54324192561393372528670455523, 7.41247013570278693973997554829, 8.583213525392232027826537768055, 9.235241798379834158610295134089
