L(s) = 1 | − 0.457·2-s − 2.69·3-s − 1.79·4-s − 3.39·5-s + 1.23·6-s + 1.04·7-s + 1.73·8-s + 4.28·9-s + 1.55·10-s + 1.34·11-s + 4.83·12-s − 5.35·13-s − 0.479·14-s + 9.17·15-s + 2.78·16-s − 1.96·17-s − 1.95·18-s + 3.39·19-s + 6.08·20-s − 2.82·21-s − 0.614·22-s − 6.44·23-s − 4.67·24-s + 6.54·25-s + 2.44·26-s − 3.46·27-s − 1.87·28-s + ⋯ |
L(s) = 1 | − 0.323·2-s − 1.55·3-s − 0.895·4-s − 1.51·5-s + 0.503·6-s + 0.396·7-s + 0.612·8-s + 1.42·9-s + 0.491·10-s + 0.405·11-s + 1.39·12-s − 1.48·13-s − 0.128·14-s + 2.36·15-s + 0.697·16-s − 0.477·17-s − 0.461·18-s + 0.777·19-s + 1.36·20-s − 0.617·21-s − 0.131·22-s − 1.34·23-s − 0.954·24-s + 1.30·25-s + 0.479·26-s − 0.666·27-s − 0.354·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2671 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.002520689041\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.002520689041\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2671 | \( 1 - T \) |
good | 2 | \( 1 + 0.457T + 2T^{2} \) |
| 3 | \( 1 + 2.69T + 3T^{2} \) |
| 5 | \( 1 + 3.39T + 5T^{2} \) |
| 7 | \( 1 - 1.04T + 7T^{2} \) |
| 11 | \( 1 - 1.34T + 11T^{2} \) |
| 13 | \( 1 + 5.35T + 13T^{2} \) |
| 17 | \( 1 + 1.96T + 17T^{2} \) |
| 19 | \( 1 - 3.39T + 19T^{2} \) |
| 23 | \( 1 + 6.44T + 23T^{2} \) |
| 29 | \( 1 + 8.28T + 29T^{2} \) |
| 31 | \( 1 + 9.12T + 31T^{2} \) |
| 37 | \( 1 + 4.38T + 37T^{2} \) |
| 41 | \( 1 + 6.24T + 41T^{2} \) |
| 43 | \( 1 + 11.1T + 43T^{2} \) |
| 47 | \( 1 - 9.37T + 47T^{2} \) |
| 53 | \( 1 + 8.06T + 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 + 3.94T + 61T^{2} \) |
| 67 | \( 1 + 5.41T + 67T^{2} \) |
| 71 | \( 1 + 1.38T + 71T^{2} \) |
| 73 | \( 1 + 5.96T + 73T^{2} \) |
| 79 | \( 1 - 0.145T + 79T^{2} \) |
| 83 | \( 1 - 6.33T + 83T^{2} \) |
| 89 | \( 1 - 15.9T + 89T^{2} \) |
| 97 | \( 1 + 9.69T + 97T^{2} \) |
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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
−8.873604102479083258934150818581, −7.85405308764334125976854267184, −7.47357311263483260561514019258, −6.69607280260439949145113036239, −5.41416282635995626634654619343, −5.05213861497230817321935735533, −4.22018072517702056926261561054, −3.60365988088319677193949432124, −1.62433093855055463189100936620, −0.03635632396009229596729626089,
0.03635632396009229596729626089, 1.62433093855055463189100936620, 3.60365988088319677193949432124, 4.22018072517702056926261561054, 5.05213861497230817321935735533, 5.41416282635995626634654619343, 6.69607280260439949145113036239, 7.47357311263483260561514019258, 7.85405308764334125976854267184, 8.873604102479083258934150818581