| L(s) = 1 | + 2.63·2-s + 4.96·4-s + 1.13·5-s + 1.99·7-s + 7.82·8-s + 3.00·10-s − 1.82·11-s + 7.15·13-s + 5.27·14-s + 10.7·16-s − 7.66·17-s + 7.80·19-s + 5.65·20-s − 4.82·22-s − 23-s − 3.70·25-s + 18.8·26-s + 9.91·28-s − 29-s − 8.74·31-s + 12.6·32-s − 20.2·34-s + 2.27·35-s + 3.85·37-s + 20.6·38-s + 8.90·40-s + 1.74·41-s + ⋯ |
| L(s) = 1 | + 1.86·2-s + 2.48·4-s + 0.508·5-s + 0.754·7-s + 2.76·8-s + 0.949·10-s − 0.550·11-s + 1.98·13-s + 1.40·14-s + 2.68·16-s − 1.85·17-s + 1.79·19-s + 1.26·20-s − 1.02·22-s − 0.208·23-s − 0.741·25-s + 3.70·26-s + 1.87·28-s − 0.185·29-s − 1.57·31-s + 2.23·32-s − 3.46·34-s + 0.384·35-s + 0.633·37-s + 3.34·38-s + 1.40·40-s + 0.272·41-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
| \(L(1)\) |
\(\approx\) |
\(8.877453730\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.877453730\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.63T + 2T^{2} \) |
| 5 | \( 1 - 1.13T + 5T^{2} \) |
| 7 | \( 1 - 1.99T + 7T^{2} \) |
| 11 | \( 1 + 1.82T + 11T^{2} \) |
| 13 | \( 1 - 7.15T + 13T^{2} \) |
| 17 | \( 1 + 7.66T + 17T^{2} \) |
| 19 | \( 1 - 7.80T + 19T^{2} \) |
| 31 | \( 1 + 8.74T + 31T^{2} \) |
| 37 | \( 1 - 3.85T + 37T^{2} \) |
| 41 | \( 1 - 1.74T + 41T^{2} \) |
| 43 | \( 1 - 7.55T + 43T^{2} \) |
| 47 | \( 1 - 12.7T + 47T^{2} \) |
| 53 | \( 1 + 0.821T + 53T^{2} \) |
| 59 | \( 1 + 1.89T + 59T^{2} \) |
| 61 | \( 1 + 6.35T + 61T^{2} \) |
| 67 | \( 1 + 3.73T + 67T^{2} \) |
| 71 | \( 1 + 7.52T + 71T^{2} \) |
| 73 | \( 1 + 11.6T + 73T^{2} \) |
| 79 | \( 1 - 10.2T + 79T^{2} \) |
| 83 | \( 1 + 3.44T + 83T^{2} \) |
| 89 | \( 1 - 3.74T + 89T^{2} \) |
| 97 | \( 1 + 12.1T + 97T^{2} \) |
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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
−7.68297158480946808255917907123, −7.24617544836590790457649126833, −6.20250170858207619908660358130, −5.83866485308396423654739934530, −5.26497109647322259279183241982, −4.37346845708551082031596118148, −3.87163281933075361864816249684, −2.97745759209332022458179182981, −2.11089912022194284217834380393, −1.37173542803778051361849090420,
1.37173542803778051361849090420, 2.11089912022194284217834380393, 2.97745759209332022458179182981, 3.87163281933075361864816249684, 4.37346845708551082031596118148, 5.26497109647322259279183241982, 5.83866485308396423654739934530, 6.20250170858207619908660358130, 7.24617544836590790457649126833, 7.68297158480946808255917907123
