L(s) = 1 | − 1.04·2-s − 2.75·3-s − 0.903·4-s + 5-s + 2.88·6-s + 3.42·7-s + 3.04·8-s + 4.61·9-s − 1.04·10-s − 2.38·11-s + 2.49·12-s − 3.58·14-s − 2.75·15-s − 1.37·16-s − 7.43·17-s − 4.82·18-s + 0.840·19-s − 0.903·20-s − 9.44·21-s + 2.50·22-s + 5.46·23-s − 8.38·24-s + 25-s − 4.44·27-s − 3.09·28-s + 1.65·29-s + 2.88·30-s + ⋯ |
L(s) = 1 | − 0.740·2-s − 1.59·3-s − 0.451·4-s + 0.447·5-s + 1.17·6-s + 1.29·7-s + 1.07·8-s + 1.53·9-s − 0.331·10-s − 0.719·11-s + 0.719·12-s − 0.958·14-s − 0.712·15-s − 0.344·16-s − 1.80·17-s − 1.13·18-s + 0.192·19-s − 0.201·20-s − 2.06·21-s + 0.533·22-s + 1.14·23-s − 1.71·24-s + 0.200·25-s − 0.854·27-s − 0.584·28-s + 0.307·29-s + 0.527·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5556384269\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5556384269\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 1.04T + 2T^{2} \) |
| 3 | \( 1 + 2.75T + 3T^{2} \) |
| 7 | \( 1 - 3.42T + 7T^{2} \) |
| 11 | \( 1 + 2.38T + 11T^{2} \) |
| 17 | \( 1 + 7.43T + 17T^{2} \) |
| 19 | \( 1 - 0.840T + 19T^{2} \) |
| 23 | \( 1 - 5.46T + 23T^{2} \) |
| 29 | \( 1 - 1.65T + 29T^{2} \) |
| 31 | \( 1 + 8.10T + 31T^{2} \) |
| 37 | \( 1 - 6.39T + 37T^{2} \) |
| 41 | \( 1 + 6.15T + 41T^{2} \) |
| 43 | \( 1 - 12.0T + 43T^{2} \) |
| 47 | \( 1 - 2.33T + 47T^{2} \) |
| 53 | \( 1 - 4.35T + 53T^{2} \) |
| 59 | \( 1 - 5.11T + 59T^{2} \) |
| 61 | \( 1 + 8.84T + 61T^{2} \) |
| 67 | \( 1 - 7.17T + 67T^{2} \) |
| 71 | \( 1 - 3.28T + 71T^{2} \) |
| 73 | \( 1 - 9.41T + 73T^{2} \) |
| 79 | \( 1 - 5.89T + 79T^{2} \) |
| 83 | \( 1 + 5.14T + 83T^{2} \) |
| 89 | \( 1 - 6.03T + 89T^{2} \) |
| 97 | \( 1 + 6.33T + 97T^{2} \) |
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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
−10.44694526779144837534089944184, −9.367932934352417537588834722572, −8.631465350031728047456158380776, −7.61725843951646788827745572798, −6.81662298072705752919840981641, −5.62278516500458573010743535994, −4.96657735130816695872810547517, −4.36503708413751762950462659971, −2.02926559953273621629397001472, −0.74686564454667274842263902915,
0.74686564454667274842263902915, 2.02926559953273621629397001472, 4.36503708413751762950462659971, 4.96657735130816695872810547517, 5.62278516500458573010743535994, 6.81662298072705752919840981641, 7.61725843951646788827745572798, 8.631465350031728047456158380776, 9.367932934352417537588834722572, 10.44694526779144837534089944184