| L(s) = 1 | − 2-s − 3-s + 4-s − 2·5-s + 6-s + 3.56·7-s − 8-s + 9-s + 2·10-s + 1.56·11-s − 12-s − 13-s − 3.56·14-s + 2·15-s + 16-s + 7.56·17-s − 18-s − 2·19-s − 2·20-s − 3.56·21-s − 1.56·22-s − 5.12·23-s + 24-s − 25-s + 26-s − 27-s + 3.56·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.894·5-s + 0.408·6-s + 1.34·7-s − 0.353·8-s + 0.333·9-s + 0.632·10-s + 0.470·11-s − 0.288·12-s − 0.277·13-s − 0.951·14-s + 0.516·15-s + 0.250·16-s + 1.83·17-s − 0.235·18-s − 0.458·19-s − 0.447·20-s − 0.777·21-s − 0.332·22-s − 1.06·23-s + 0.204·24-s − 0.200·25-s + 0.196·26-s − 0.192·27-s + 0.673·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
| good | 5 | \( 1 + 2T + 5T^{2} \) |
| 7 | \( 1 - 3.56T + 7T^{2} \) |
| 11 | \( 1 - 1.56T + 11T^{2} \) |
| 17 | \( 1 - 7.56T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 + 5.12T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 7.12T + 31T^{2} \) |
| 37 | \( 1 + 0.876T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 7.12T + 43T^{2} \) |
| 47 | \( 1 - 6.24T + 47T^{2} \) |
| 53 | \( 1 - 2.43T + 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 - 2.87T + 61T^{2} \) |
| 67 | \( 1 + 0.684T + 67T^{2} \) |
| 71 | \( 1 + 14.2T + 71T^{2} \) |
| 73 | \( 1 + 8.68T + 73T^{2} \) |
| 79 | \( 1 + 17.3T + 79T^{2} \) |
| 83 | \( 1 + 13.3T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 + 12.2T + 97T^{2} \) |
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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
−7.44612128071544665986765105257, −7.20480031889648990049588348652, −6.01634244011360865153011104285, −5.54789648427194112280809942946, −4.60151109825343086580730462966, −4.02940192310934551562604174216, −3.10018059791519605389740493651, −1.84068298358037682276259880991, −1.18486051978851374158908330869, 0,
1.18486051978851374158908330869, 1.84068298358037682276259880991, 3.10018059791519605389740493651, 4.02940192310934551562604174216, 4.60151109825343086580730462966, 5.54789648427194112280809942946, 6.01634244011360865153011104285, 7.20480031889648990049588348652, 7.44612128071544665986765105257
