| L(s) = 1 | − 0.150·2-s − 3-s − 1.97·4-s + 2.78·5-s + 0.150·6-s + 2.48·7-s + 0.598·8-s + 9-s − 0.418·10-s − 0.800·11-s + 1.97·12-s − 13-s − 0.373·14-s − 2.78·15-s + 3.86·16-s + 2.39·17-s − 0.150·18-s − 6.04·19-s − 5.49·20-s − 2.48·21-s + 0.120·22-s − 3.64·23-s − 0.598·24-s + 2.73·25-s + 0.150·26-s − 27-s − 4.91·28-s + ⋯ |
| L(s) = 1 | − 0.106·2-s − 0.577·3-s − 0.988·4-s + 1.24·5-s + 0.0614·6-s + 0.939·7-s + 0.211·8-s + 0.333·9-s − 0.132·10-s − 0.241·11-s + 0.570·12-s − 0.277·13-s − 0.0999·14-s − 0.718·15-s + 0.966·16-s + 0.581·17-s − 0.0354·18-s − 1.38·19-s − 1.22·20-s − 0.542·21-s + 0.0256·22-s − 0.760·23-s − 0.122·24-s + 0.546·25-s + 0.0295·26-s − 0.192·27-s − 0.928·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
| good | 2 | \( 1 + 0.150T + 2T^{2} \) |
| 5 | \( 1 - 2.78T + 5T^{2} \) |
| 7 | \( 1 - 2.48T + 7T^{2} \) |
| 11 | \( 1 + 0.800T + 11T^{2} \) |
| 17 | \( 1 - 2.39T + 17T^{2} \) |
| 19 | \( 1 + 6.04T + 19T^{2} \) |
| 23 | \( 1 + 3.64T + 23T^{2} \) |
| 29 | \( 1 + 1.59T + 29T^{2} \) |
| 31 | \( 1 + 8.28T + 31T^{2} \) |
| 37 | \( 1 - 6.92T + 37T^{2} \) |
| 41 | \( 1 + 0.575T + 41T^{2} \) |
| 43 | \( 1 + 6.01T + 43T^{2} \) |
| 47 | \( 1 + 11.0T + 47T^{2} \) |
| 53 | \( 1 - 6.40T + 53T^{2} \) |
| 59 | \( 1 + 13.8T + 59T^{2} \) |
| 61 | \( 1 + 5.95T + 61T^{2} \) |
| 67 | \( 1 - 0.385T + 67T^{2} \) |
| 71 | \( 1 - 4.28T + 71T^{2} \) |
| 73 | \( 1 - 6.17T + 73T^{2} \) |
| 79 | \( 1 + 15.3T + 79T^{2} \) |
| 83 | \( 1 - 7.76T + 83T^{2} \) |
| 89 | \( 1 - 10.5T + 89T^{2} \) |
| 97 | \( 1 - 9.30T + 97T^{2} \) |
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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
−8.110071249833819502152851407564, −7.51178487695762038662519336438, −6.31360471667992809849797912127, −5.81242889867112098512218413610, −5.04581145601290068390791012701, −4.58560194201185411601011650889, −3.56876869201148868630058529424, −2.12936295077849628414009199867, −1.45976488432312529967511625898, 0,
1.45976488432312529967511625898, 2.12936295077849628414009199867, 3.56876869201148868630058529424, 4.58560194201185411601011650889, 5.04581145601290068390791012701, 5.81242889867112098512218413610, 6.31360471667992809849797912127, 7.51178487695762038662519336438, 8.110071249833819502152851407564
