L(s) = 1 | − 1.43·2-s + 2.32·3-s + 0.0487·4-s + 1.62·5-s − 3.33·6-s + 3.43·7-s + 2.79·8-s + 2.42·9-s − 2.32·10-s − 0.695·11-s + 0.113·12-s + 2.78·13-s − 4.91·14-s + 3.77·15-s − 4.09·16-s − 6.87·17-s − 3.46·18-s − 1.23·19-s + 0.0790·20-s + 7.99·21-s + 0.995·22-s + 8.03·23-s + 6.50·24-s − 2.37·25-s − 3.98·26-s − 1.34·27-s + 0.167·28-s + ⋯ |
L(s) = 1 | − 1.01·2-s + 1.34·3-s + 0.0243·4-s + 0.725·5-s − 1.36·6-s + 1.29·7-s + 0.987·8-s + 0.806·9-s − 0.733·10-s − 0.209·11-s + 0.0327·12-s + 0.772·13-s − 1.31·14-s + 0.974·15-s − 1.02·16-s − 1.66·17-s − 0.816·18-s − 0.282·19-s + 0.0176·20-s + 1.74·21-s + 0.212·22-s + 1.67·23-s + 1.32·24-s − 0.474·25-s − 0.782·26-s − 0.259·27-s + 0.0316·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.539510113\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.539510113\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 503 | \( 1 - T \) |
good | 2 | \( 1 + 1.43T + 2T^{2} \) |
| 3 | \( 1 - 2.32T + 3T^{2} \) |
| 5 | \( 1 - 1.62T + 5T^{2} \) |
| 7 | \( 1 - 3.43T + 7T^{2} \) |
| 11 | \( 1 + 0.695T + 11T^{2} \) |
| 13 | \( 1 - 2.78T + 13T^{2} \) |
| 17 | \( 1 + 6.87T + 17T^{2} \) |
| 19 | \( 1 + 1.23T + 19T^{2} \) |
| 23 | \( 1 - 8.03T + 23T^{2} \) |
| 29 | \( 1 + 8.02T + 29T^{2} \) |
| 31 | \( 1 - 9.59T + 31T^{2} \) |
| 37 | \( 1 + 0.0454T + 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 + 10.6T + 43T^{2} \) |
| 47 | \( 1 - 1.72T + 47T^{2} \) |
| 53 | \( 1 + 6.66T + 53T^{2} \) |
| 59 | \( 1 + 7.83T + 59T^{2} \) |
| 61 | \( 1 - 8.48T + 61T^{2} \) |
| 67 | \( 1 - 0.128T + 67T^{2} \) |
| 71 | \( 1 - 9.36T + 71T^{2} \) |
| 73 | \( 1 - 11.6T + 73T^{2} \) |
| 79 | \( 1 - 9.03T + 79T^{2} \) |
| 83 | \( 1 + 0.939T + 83T^{2} \) |
| 89 | \( 1 + 16.8T + 89T^{2} \) |
| 97 | \( 1 - 10.0T + 97T^{2} \) |
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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
−10.85476611607895665169626607505, −9.591735838233713224268900275092, −9.085031003426995824724393530555, −8.347662195588282184829153474839, −7.87136460936461021509207125846, −6.68807324606357218789201863886, −5.10986974185637095655779923254, −4.09630177157712334487693746332, −2.43794511155899922805173577606, −1.52601554781974830404929905906,
1.52601554781974830404929905906, 2.43794511155899922805173577606, 4.09630177157712334487693746332, 5.10986974185637095655779923254, 6.68807324606357218789201863886, 7.87136460936461021509207125846, 8.347662195588282184829153474839, 9.085031003426995824724393530555, 9.591735838233713224268900275092, 10.85476611607895665169626607505