L(s) = 1 | − 0.503·2-s − 3.01·3-s − 1.74·4-s + 5-s + 1.51·6-s + 2.48·7-s + 1.88·8-s + 6.06·9-s − 0.503·10-s + 4.58·11-s + 5.25·12-s + 6.06·13-s − 1.25·14-s − 3.01·15-s + 2.54·16-s + 1.83·17-s − 3.05·18-s − 1.74·20-s − 7.49·21-s − 2.30·22-s + 0.125·23-s − 5.67·24-s + 25-s − 3.05·26-s − 9.21·27-s − 4.34·28-s − 3.79·29-s + ⋯ |
L(s) = 1 | − 0.356·2-s − 1.73·3-s − 0.873·4-s + 0.447·5-s + 0.618·6-s + 0.941·7-s + 0.666·8-s + 2.02·9-s − 0.159·10-s + 1.38·11-s + 1.51·12-s + 1.68·13-s − 0.335·14-s − 0.777·15-s + 0.635·16-s + 0.444·17-s − 0.719·18-s − 0.390·20-s − 1.63·21-s − 0.491·22-s + 0.0262·23-s − 1.15·24-s + 0.200·25-s − 0.598·26-s − 1.77·27-s − 0.821·28-s − 0.703·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.020390655\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.020390655\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + 0.503T + 2T^{2} \) |
| 3 | \( 1 + 3.01T + 3T^{2} \) |
| 7 | \( 1 - 2.48T + 7T^{2} \) |
| 11 | \( 1 - 4.58T + 11T^{2} \) |
| 13 | \( 1 - 6.06T + 13T^{2} \) |
| 17 | \( 1 - 1.83T + 17T^{2} \) |
| 23 | \( 1 - 0.125T + 23T^{2} \) |
| 29 | \( 1 + 3.79T + 29T^{2} \) |
| 31 | \( 1 - 8.09T + 31T^{2} \) |
| 37 | \( 1 + 1.44T + 37T^{2} \) |
| 41 | \( 1 + 3.02T + 41T^{2} \) |
| 43 | \( 1 - 8.78T + 43T^{2} \) |
| 47 | \( 1 - 5.70T + 47T^{2} \) |
| 53 | \( 1 - 9.24T + 53T^{2} \) |
| 59 | \( 1 + 7.06T + 59T^{2} \) |
| 61 | \( 1 + 8.41T + 61T^{2} \) |
| 67 | \( 1 - 6.08T + 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 + 10.8T + 73T^{2} \) |
| 79 | \( 1 - 5.23T + 79T^{2} \) |
| 83 | \( 1 - 5.03T + 83T^{2} \) |
| 89 | \( 1 + 14.6T + 89T^{2} \) |
| 97 | \( 1 + 2.49T + 97T^{2} \) |
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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
−9.289772796560999034230062174094, −8.636505754424566468860330425190, −7.71019839873704282763957078713, −6.63958328658564291861369565005, −5.97213569413620777241214963554, −5.33657565508549433675573515076, −4.44413807339140922425978076742, −3.83171457425077008087788177568, −1.45777694367893897722316644020, −0.964360186067930435772160390631,
0.964360186067930435772160390631, 1.45777694367893897722316644020, 3.83171457425077008087788177568, 4.44413807339140922425978076742, 5.33657565508549433675573515076, 5.97213569413620777241214963554, 6.63958328658564291861369565005, 7.71019839873704282763957078713, 8.636505754424566468860330425190, 9.289772796560999034230062174094