| L(s) = 1 | + 2-s − 1.98·3-s + 4-s + 5-s − 1.98·6-s + 2.71·7-s + 8-s + 0.946·9-s + 10-s + 11-s − 1.98·12-s + 5.23·13-s + 2.71·14-s − 1.98·15-s + 16-s − 4.62·17-s + 0.946·18-s + 4.03·19-s + 20-s − 5.39·21-s + 22-s − 1.87·23-s − 1.98·24-s + 25-s + 5.23·26-s + 4.07·27-s + 2.71·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.14·3-s + 0.5·4-s + 0.447·5-s − 0.811·6-s + 1.02·7-s + 0.353·8-s + 0.315·9-s + 0.316·10-s + 0.301·11-s − 0.573·12-s + 1.45·13-s + 0.726·14-s − 0.512·15-s + 0.250·16-s − 1.12·17-s + 0.223·18-s + 0.925·19-s + 0.223·20-s − 1.17·21-s + 0.213·22-s − 0.391·23-s − 0.405·24-s + 0.200·25-s + 1.02·26-s + 0.785·27-s + 0.513·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.835263701\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.835263701\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| good | 3 | \( 1 + 1.98T + 3T^{2} \) |
| 7 | \( 1 - 2.71T + 7T^{2} \) |
| 13 | \( 1 - 5.23T + 13T^{2} \) |
| 17 | \( 1 + 4.62T + 17T^{2} \) |
| 19 | \( 1 - 4.03T + 19T^{2} \) |
| 23 | \( 1 + 1.87T + 23T^{2} \) |
| 29 | \( 1 - 1.51T + 29T^{2} \) |
| 31 | \( 1 - 7.16T + 31T^{2} \) |
| 37 | \( 1 - 6.02T + 37T^{2} \) |
| 41 | \( 1 + 12.1T + 41T^{2} \) |
| 47 | \( 1 + 3.84T + 47T^{2} \) |
| 53 | \( 1 - 10.6T + 53T^{2} \) |
| 59 | \( 1 - 9.84T + 59T^{2} \) |
| 61 | \( 1 - 5.53T + 61T^{2} \) |
| 67 | \( 1 + 3.05T + 67T^{2} \) |
| 71 | \( 1 + 14.0T + 71T^{2} \) |
| 73 | \( 1 + 15.9T + 73T^{2} \) |
| 79 | \( 1 - 1.06T + 79T^{2} \) |
| 83 | \( 1 - 0.664T + 83T^{2} \) |
| 89 | \( 1 - 5.09T + 89T^{2} \) |
| 97 | \( 1 - 15.4T + 97T^{2} \) |
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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
−8.409268364951199504460652900099, −7.30805018197826530593936875122, −6.47292907787538501575152780243, −6.09760791443432081679970394784, −5.33932194201795865260899738015, −4.75161463627795261987844097104, −4.04660582460390441790366298180, −2.95373770833378135054018169138, −1.79459521336370474249015982742, −0.953626228133436868857837721851,
0.953626228133436868857837721851, 1.79459521336370474249015982742, 2.95373770833378135054018169138, 4.04660582460390441790366298180, 4.75161463627795261987844097104, 5.33932194201795865260899738015, 6.09760791443432081679970394784, 6.47292907787538501575152780243, 7.30805018197826530593936875122, 8.409268364951199504460652900099
