| L(s) = 1 | + 2.42·2-s + 3.87·4-s + 0.275·5-s − 5.16·7-s + 4.55·8-s + 0.667·10-s + 4.59·11-s + 2.58·13-s − 12.5·14-s + 3.27·16-s − 6.77·17-s − 7.93·19-s + 1.06·20-s + 11.1·22-s − 23-s − 4.92·25-s + 6.26·26-s − 20.0·28-s + 29-s − 1.54·31-s − 1.15·32-s − 16.4·34-s − 1.42·35-s − 1.24·37-s − 19.2·38-s + 1.25·40-s + 1.39·41-s + ⋯ |
| L(s) = 1 | + 1.71·2-s + 1.93·4-s + 0.123·5-s − 1.95·7-s + 1.60·8-s + 0.210·10-s + 1.38·11-s + 0.716·13-s − 3.34·14-s + 0.819·16-s − 1.64·17-s − 1.82·19-s + 0.238·20-s + 2.37·22-s − 0.208·23-s − 0.984·25-s + 1.22·26-s − 3.78·28-s + 0.185·29-s − 0.277·31-s − 0.204·32-s − 2.81·34-s − 0.240·35-s − 0.204·37-s − 3.12·38-s + 0.198·40-s + 0.218·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 2.42T + 2T^{2} \) |
| 5 | \( 1 - 0.275T + 5T^{2} \) |
| 7 | \( 1 + 5.16T + 7T^{2} \) |
| 11 | \( 1 - 4.59T + 11T^{2} \) |
| 13 | \( 1 - 2.58T + 13T^{2} \) |
| 17 | \( 1 + 6.77T + 17T^{2} \) |
| 19 | \( 1 + 7.93T + 19T^{2} \) |
| 31 | \( 1 + 1.54T + 31T^{2} \) |
| 37 | \( 1 + 1.24T + 37T^{2} \) |
| 41 | \( 1 - 1.39T + 41T^{2} \) |
| 43 | \( 1 + 0.777T + 43T^{2} \) |
| 47 | \( 1 - 5.57T + 47T^{2} \) |
| 53 | \( 1 + 11.5T + 53T^{2} \) |
| 59 | \( 1 + 7.31T + 59T^{2} \) |
| 61 | \( 1 - 1.28T + 61T^{2} \) |
| 67 | \( 1 - 10.1T + 67T^{2} \) |
| 71 | \( 1 + 4.16T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 + 1.19T + 79T^{2} \) |
| 83 | \( 1 - 5.53T + 83T^{2} \) |
| 89 | \( 1 + 13.4T + 89T^{2} \) |
| 97 | \( 1 - 3.41T + 97T^{2} \) |
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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
−7.13172134689178289476048622959, −6.60142495255435943393398473257, −6.21495758232297484687916717931, −5.86891214669243105449204385864, −4.50747978217142212996645377004, −4.03243848623366248886420091220, −3.53921077531780806070947159132, −2.63818064822826416518095111742, −1.83360709670222652562329926573, 0,
1.83360709670222652562329926573, 2.63818064822826416518095111742, 3.53921077531780806070947159132, 4.03243848623366248886420091220, 4.50747978217142212996645377004, 5.86891214669243105449204385864, 6.21495758232297484687916717931, 6.60142495255435943393398473257, 7.13172134689178289476048622959
