L(s) = 1 | + 2·2-s − 7.16·3-s + 4·4-s − 5·5-s − 14.3·6-s − 25.3·7-s + 8·8-s + 24.3·9-s − 10·10-s + 67.0·11-s − 28.6·12-s − 2.41·13-s − 50.7·14-s + 35.8·15-s + 16·16-s − 12.5·17-s + 48.7·18-s + 104.·19-s − 20·20-s + 182.·21-s + 134.·22-s + 23·23-s − 57.3·24-s + 25·25-s − 4.83·26-s + 18.6·27-s − 101.·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.37·3-s + 0.5·4-s − 0.447·5-s − 0.975·6-s − 1.37·7-s + 0.353·8-s + 0.903·9-s − 0.316·10-s + 1.83·11-s − 0.689·12-s − 0.0515·13-s − 0.969·14-s + 0.617·15-s + 0.250·16-s − 0.178·17-s + 0.638·18-s + 1.26·19-s − 0.223·20-s + 1.89·21-s + 1.29·22-s + 0.208·23-s − 0.487·24-s + 0.200·25-s − 0.0364·26-s + 0.132·27-s − 0.685·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.419787515\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.419787515\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 2T \) |
| 5 | \( 1 + 5T \) |
| 23 | \( 1 - 23T \) |
good | 3 | \( 1 + 7.16T + 27T^{2} \) |
| 7 | \( 1 + 25.3T + 343T^{2} \) |
| 11 | \( 1 - 67.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 2.41T + 2.19e3T^{2} \) |
| 17 | \( 1 + 12.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 104.T + 6.85e3T^{2} \) |
| 29 | \( 1 - 221.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 102.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 2.56T + 5.06e4T^{2} \) |
| 41 | \( 1 + 89.4T + 6.89e4T^{2} \) |
| 43 | \( 1 - 5.94T + 7.95e4T^{2} \) |
| 47 | \( 1 - 549.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 159.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 593.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 894.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 525.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 57.4T + 3.57e5T^{2} \) |
| 73 | \( 1 + 870.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 578.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 345.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 311.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.81e3T + 9.12e5T^{2} \) |
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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
−12.05233486089650901785050739248, −11.13641895685155115962222837489, −10.06170260565568284017403427533, −9.023958072750168503784701341859, −7.11143538519143037905885860852, −6.51236104036973715589930832897, −5.66306889719802068330073705935, −4.36306223199127860417851480160, −3.28142027554712280462588701528, −0.867854554964773391357946850720,
0.867854554964773391357946850720, 3.28142027554712280462588701528, 4.36306223199127860417851480160, 5.66306889719802068330073705935, 6.51236104036973715589930832897, 7.11143538519143037905885860852, 9.023958072750168503784701341859, 10.06170260565568284017403427533, 11.13641895685155115962222837489, 12.05233486089650901785050739248