L(s) = 1 | − 2.69·2-s + 3.13·3-s + 5.25·4-s + 0.629·5-s − 8.43·6-s − 1.84·7-s − 8.75·8-s + 6.81·9-s − 1.69·10-s − 11-s + 16.4·12-s + 0.191·13-s + 4.97·14-s + 1.97·15-s + 13.0·16-s + 1.18·17-s − 18.3·18-s + 4.49·19-s + 3.30·20-s − 5.78·21-s + 2.69·22-s + 1.31·23-s − 27.4·24-s − 4.60·25-s − 0.515·26-s + 11.9·27-s − 9.69·28-s + ⋯ |
L(s) = 1 | − 1.90·2-s + 1.80·3-s + 2.62·4-s + 0.281·5-s − 3.44·6-s − 0.697·7-s − 3.09·8-s + 2.27·9-s − 0.536·10-s − 0.301·11-s + 4.74·12-s + 0.0530·13-s + 1.32·14-s + 0.509·15-s + 3.26·16-s + 0.286·17-s − 4.32·18-s + 1.03·19-s + 0.739·20-s − 1.26·21-s + 0.574·22-s + 0.273·23-s − 5.59·24-s − 0.920·25-s − 0.101·26-s + 2.30·27-s − 1.83·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.718864233\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.718864233\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + T \) |
| 547 | \( 1 - T \) |
good | 2 | \( 1 + 2.69T + 2T^{2} \) |
| 3 | \( 1 - 3.13T + 3T^{2} \) |
| 5 | \( 1 - 0.629T + 5T^{2} \) |
| 7 | \( 1 + 1.84T + 7T^{2} \) |
| 13 | \( 1 - 0.191T + 13T^{2} \) |
| 17 | \( 1 - 1.18T + 17T^{2} \) |
| 19 | \( 1 - 4.49T + 19T^{2} \) |
| 23 | \( 1 - 1.31T + 23T^{2} \) |
| 29 | \( 1 + 5.08T + 29T^{2} \) |
| 31 | \( 1 - 0.491T + 31T^{2} \) |
| 37 | \( 1 - 8.46T + 37T^{2} \) |
| 41 | \( 1 - 5.11T + 41T^{2} \) |
| 43 | \( 1 + 3.80T + 43T^{2} \) |
| 47 | \( 1 - 7.13T + 47T^{2} \) |
| 53 | \( 1 + 4.73T + 53T^{2} \) |
| 59 | \( 1 - 8.59T + 59T^{2} \) |
| 61 | \( 1 - 9.02T + 61T^{2} \) |
| 67 | \( 1 - 5.58T + 67T^{2} \) |
| 71 | \( 1 + 2.00T + 71T^{2} \) |
| 73 | \( 1 - 0.552T + 73T^{2} \) |
| 79 | \( 1 - 9.93T + 79T^{2} \) |
| 83 | \( 1 - 5.01T + 83T^{2} \) |
| 89 | \( 1 + 12.3T + 89T^{2} \) |
| 97 | \( 1 - 16.3T + 97T^{2} \) |
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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
−8.030385191535685303868923595623, −7.78328080484417844856312418340, −7.14206968551287567095238506260, −6.43292812767026030272394807847, −5.49850132921010326503300083921, −3.92334571159486060738018495285, −3.16267339469695040182113386799, −2.53458923572810864376995546683, −1.85207650618670305723180173274, −0.843474778415633974455411699579,
0.843474778415633974455411699579, 1.85207650618670305723180173274, 2.53458923572810864376995546683, 3.16267339469695040182113386799, 3.92334571159486060738018495285, 5.49850132921010326503300083921, 6.43292812767026030272394807847, 7.14206968551287567095238506260, 7.78328080484417844856312418340, 8.030385191535685303868923595623