L(s) = 1 | − 2.60·2-s − 2.10·3-s + 4.77·4-s − 3.11·5-s + 5.48·6-s − 2.87·7-s − 7.23·8-s + 1.43·9-s + 8.11·10-s + 2.41·11-s − 10.0·12-s + 1.83·13-s + 7.48·14-s + 6.56·15-s + 9.27·16-s − 1.14·17-s − 3.73·18-s − 2.71·19-s − 14.8·20-s + 6.05·21-s − 6.29·22-s + 7.41·23-s + 15.2·24-s + 4.72·25-s − 4.77·26-s + 3.29·27-s − 13.7·28-s + ⋯ |
L(s) = 1 | − 1.84·2-s − 1.21·3-s + 2.38·4-s − 1.39·5-s + 2.23·6-s − 1.08·7-s − 2.55·8-s + 0.477·9-s + 2.56·10-s + 0.729·11-s − 2.90·12-s + 0.508·13-s + 2.00·14-s + 1.69·15-s + 2.31·16-s − 0.278·17-s − 0.879·18-s − 0.623·19-s − 3.33·20-s + 1.32·21-s − 1.34·22-s + 1.54·23-s + 3.10·24-s + 0.944·25-s − 0.936·26-s + 0.634·27-s − 2.59·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2194568801\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2194568801\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 53 | \( 1 + T \) |
| 151 | \( 1 - T \) |
good | 2 | \( 1 + 2.60T + 2T^{2} \) |
| 3 | \( 1 + 2.10T + 3T^{2} \) |
| 5 | \( 1 + 3.11T + 5T^{2} \) |
| 7 | \( 1 + 2.87T + 7T^{2} \) |
| 11 | \( 1 - 2.41T + 11T^{2} \) |
| 13 | \( 1 - 1.83T + 13T^{2} \) |
| 17 | \( 1 + 1.14T + 17T^{2} \) |
| 19 | \( 1 + 2.71T + 19T^{2} \) |
| 23 | \( 1 - 7.41T + 23T^{2} \) |
| 29 | \( 1 + 8.43T + 29T^{2} \) |
| 31 | \( 1 + 3.35T + 31T^{2} \) |
| 37 | \( 1 - 0.173T + 37T^{2} \) |
| 41 | \( 1 - 5.90T + 41T^{2} \) |
| 43 | \( 1 - 11.2T + 43T^{2} \) |
| 47 | \( 1 - 8.83T + 47T^{2} \) |
| 59 | \( 1 + 11.1T + 59T^{2} \) |
| 61 | \( 1 - 4.60T + 61T^{2} \) |
| 67 | \( 1 - 11.9T + 67T^{2} \) |
| 71 | \( 1 - 5.66T + 71T^{2} \) |
| 73 | \( 1 - 7.28T + 73T^{2} \) |
| 79 | \( 1 - 17.2T + 79T^{2} \) |
| 83 | \( 1 - 6.58T + 83T^{2} \) |
| 89 | \( 1 + 9.21T + 89T^{2} \) |
| 97 | \( 1 - 11.4T + 97T^{2} \) |
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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
−7.70937044775614011361090475212, −7.31225097899619440124074999906, −6.54685597178922827250924896205, −6.26896528408804896569028094448, −5.31840366321432173526404224936, −4.09595753668373888943072996746, −3.45931026157720908976050208085, −2.41201735207766641848646484740, −1.03783901187150578992124145625, −0.41568501936849489672831377305,
0.41568501936849489672831377305, 1.03783901187150578992124145625, 2.41201735207766641848646484740, 3.45931026157720908976050208085, 4.09595753668373888943072996746, 5.31840366321432173526404224936, 6.26896528408804896569028094448, 6.54685597178922827250924896205, 7.31225097899619440124074999906, 7.70937044775614011361090475212