| L(s) = 1 | − 2.00·2-s + 2.01·4-s − 3.94·5-s − 2.43·7-s − 0.0387·8-s + 7.90·10-s − 5.06·11-s + 0.849·13-s + 4.88·14-s − 3.96·16-s + 3.54·17-s + 2.59·19-s − 7.96·20-s + 10.1·22-s + 23-s + 10.5·25-s − 1.70·26-s − 4.92·28-s + 29-s − 0.555·31-s + 8.01·32-s − 7.10·34-s + 9.61·35-s − 2.47·37-s − 5.19·38-s + 0.152·40-s − 2.45·41-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.00·4-s − 1.76·5-s − 0.921·7-s − 0.0136·8-s + 2.50·10-s − 1.52·11-s + 0.235·13-s + 1.30·14-s − 0.990·16-s + 0.859·17-s + 0.594·19-s − 1.78·20-s + 2.16·22-s + 0.208·23-s + 2.11·25-s − 0.334·26-s − 0.929·28-s + 0.185·29-s − 0.0998·31-s + 1.41·32-s − 1.21·34-s + 1.62·35-s − 0.407·37-s − 0.842·38-s + 0.0241·40-s − 0.383·41-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
| \(L(1)\) |
\(\approx\) |
\(0.08855874390\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08855874390\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.00T + 2T^{2} \) |
| 5 | \( 1 + 3.94T + 5T^{2} \) |
| 7 | \( 1 + 2.43T + 7T^{2} \) |
| 11 | \( 1 + 5.06T + 11T^{2} \) |
| 13 | \( 1 - 0.849T + 13T^{2} \) |
| 17 | \( 1 - 3.54T + 17T^{2} \) |
| 19 | \( 1 - 2.59T + 19T^{2} \) |
| 31 | \( 1 + 0.555T + 31T^{2} \) |
| 37 | \( 1 + 2.47T + 37T^{2} \) |
| 41 | \( 1 + 2.45T + 41T^{2} \) |
| 43 | \( 1 + 7.48T + 43T^{2} \) |
| 47 | \( 1 + 4.02T + 47T^{2} \) |
| 53 | \( 1 + 3.73T + 53T^{2} \) |
| 59 | \( 1 + 9.66T + 59T^{2} \) |
| 61 | \( 1 + 6.98T + 61T^{2} \) |
| 67 | \( 1 + 13.0T + 67T^{2} \) |
| 71 | \( 1 + 6.82T + 71T^{2} \) |
| 73 | \( 1 + 15.2T + 73T^{2} \) |
| 79 | \( 1 - 8.57T + 79T^{2} \) |
| 83 | \( 1 + 8.09T + 83T^{2} \) |
| 89 | \( 1 + 3.09T + 89T^{2} \) |
| 97 | \( 1 - 0.381T + 97T^{2} \) |
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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
−8.080490892506603383385486992011, −7.54709874213970793535363787927, −7.18415224874923492461382695790, −6.24733501166138651275679052766, −5.13889969986335538838073563799, −4.42113224257857833549563200668, −3.27334235304976010945204250145, −2.97240986930620852204434066455, −1.41084550519628252840304639903, −0.20339993281210689979117329034,
0.20339993281210689979117329034, 1.41084550519628252840304639903, 2.97240986930620852204434066455, 3.27334235304976010945204250145, 4.42113224257857833549563200668, 5.13889969986335538838073563799, 6.24733501166138651275679052766, 7.18415224874923492461382695790, 7.54709874213970793535363787927, 8.080490892506603383385486992011
