L(s) = 1 | + 2-s − 2.46·3-s + 4-s + 2.99·5-s − 2.46·6-s + 1.50·7-s + 8-s + 3.06·9-s + 2.99·10-s + 5.08·11-s − 2.46·12-s + 2.98·13-s + 1.50·14-s − 7.37·15-s + 16-s + 2.94·17-s + 3.06·18-s + 6.04·19-s + 2.99·20-s − 3.70·21-s + 5.08·22-s − 4.72·23-s − 2.46·24-s + 3.97·25-s + 2.98·26-s − 0.150·27-s + 1.50·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.42·3-s + 0.5·4-s + 1.33·5-s − 1.00·6-s + 0.569·7-s + 0.353·8-s + 1.02·9-s + 0.947·10-s + 1.53·11-s − 0.710·12-s + 0.828·13-s + 0.402·14-s − 1.90·15-s + 0.250·16-s + 0.715·17-s + 0.721·18-s + 1.38·19-s + 0.669·20-s − 0.808·21-s + 1.08·22-s − 0.986·23-s − 0.502·24-s + 0.795·25-s + 0.585·26-s − 0.0288·27-s + 0.284·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.261886223\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.261886223\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 2011 | \( 1 + T \) |
good | 3 | \( 1 + 2.46T + 3T^{2} \) |
| 5 | \( 1 - 2.99T + 5T^{2} \) |
| 7 | \( 1 - 1.50T + 7T^{2} \) |
| 11 | \( 1 - 5.08T + 11T^{2} \) |
| 13 | \( 1 - 2.98T + 13T^{2} \) |
| 17 | \( 1 - 2.94T + 17T^{2} \) |
| 19 | \( 1 - 6.04T + 19T^{2} \) |
| 23 | \( 1 + 4.72T + 23T^{2} \) |
| 29 | \( 1 + 3.65T + 29T^{2} \) |
| 31 | \( 1 - 6.39T + 31T^{2} \) |
| 37 | \( 1 - 7.83T + 37T^{2} \) |
| 41 | \( 1 + 7.97T + 41T^{2} \) |
| 43 | \( 1 - 6.69T + 43T^{2} \) |
| 47 | \( 1 + 9.45T + 47T^{2} \) |
| 53 | \( 1 + 0.721T + 53T^{2} \) |
| 59 | \( 1 - 3.68T + 59T^{2} \) |
| 61 | \( 1 + 8.85T + 61T^{2} \) |
| 67 | \( 1 + 3.85T + 67T^{2} \) |
| 71 | \( 1 + 11.5T + 71T^{2} \) |
| 73 | \( 1 + 11.0T + 73T^{2} \) |
| 79 | \( 1 - 10.8T + 79T^{2} \) |
| 83 | \( 1 + 4.50T + 83T^{2} \) |
| 89 | \( 1 + 11.7T + 89T^{2} \) |
| 97 | \( 1 - 3.71T + 97T^{2} \) |
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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
−8.398089226273601166127045625919, −7.41194838173875414263341571337, −6.47930159299903967082253856326, −6.07313078068868337656388101288, −5.64452143929536735662025385426, −4.88594560766608434544532849029, −4.11093363042755401701045822100, −3.06312256448385925177170621166, −1.61172755849858524677707769191, −1.19440951133912059127237391269,
1.19440951133912059127237391269, 1.61172755849858524677707769191, 3.06312256448385925177170621166, 4.11093363042755401701045822100, 4.88594560766608434544532849029, 5.64452143929536735662025385426, 6.07313078068868337656388101288, 6.47930159299903967082253856326, 7.41194838173875414263341571337, 8.398089226273601166127045625919