L(s) = 1 | + (0.186 + 1.29i)3-s + (0.415 + 0.909i)4-s + (−1.61 − 0.474i)5-s + (−0.686 + 0.201i)9-s + (0.841 − 0.540i)11-s + (−1.10 + 0.708i)12-s + (0.313 − 2.18i)15-s + (−0.654 + 0.755i)16-s + (−0.239 − 1.66i)20-s + (0.841 − 0.540i)23-s + (1.54 + 0.989i)25-s + (0.154 + 0.339i)27-s + (0.273 − 1.89i)31-s + (0.857 + 0.989i)33-s + (−0.468 − 0.540i)36-s + (−0.797 + 0.234i)37-s + ⋯ |
L(s) = 1 | + (0.186 + 1.29i)3-s + (0.415 + 0.909i)4-s + (−1.61 − 0.474i)5-s + (−0.686 + 0.201i)9-s + (0.841 − 0.540i)11-s + (−1.10 + 0.708i)12-s + (0.313 − 2.18i)15-s + (−0.654 + 0.755i)16-s + (−0.239 − 1.66i)20-s + (0.841 − 0.540i)23-s + (1.54 + 0.989i)25-s + (0.154 + 0.339i)27-s + (0.273 − 1.89i)31-s + (0.857 + 0.989i)33-s + (−0.468 − 0.540i)36-s + (−0.797 + 0.234i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.102 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.102 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7027039520\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7027039520\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + (-0.841 + 0.540i)T \) |
| 23 | \( 1 + (-0.841 + 0.540i)T \) |
good | 2 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 3 | \( 1 + (-0.186 - 1.29i)T + (-0.959 + 0.281i)T^{2} \) |
| 5 | \( 1 + (1.61 + 0.474i)T + (0.841 + 0.540i)T^{2} \) |
| 7 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 13 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 17 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 19 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 29 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 31 | \( 1 + (-0.273 + 1.89i)T + (-0.959 - 0.281i)T^{2} \) |
| 37 | \( 1 + (0.797 - 0.234i)T + (0.841 - 0.540i)T^{2} \) |
| 41 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 43 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 47 | \( 1 + 1.91T + T^{2} \) |
| 53 | \( 1 + (0.544 - 0.627i)T + (-0.142 - 0.989i)T^{2} \) |
| 59 | \( 1 + (0.544 + 0.627i)T + (-0.142 + 0.989i)T^{2} \) |
| 61 | \( 1 + (0.959 + 0.281i)T^{2} \) |
| 67 | \( 1 + (1.10 + 0.708i)T + (0.415 + 0.909i)T^{2} \) |
| 71 | \( 1 + (-1.41 - 0.909i)T + (0.415 + 0.909i)T^{2} \) |
| 73 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 79 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 83 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 89 | \( 1 + (-0.186 - 1.29i)T + (-0.959 + 0.281i)T^{2} \) |
| 97 | \( 1 + (-1.84 - 0.540i)T + (0.841 + 0.540i)T^{2} \) |
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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
−12.20502622489614933364795281466, −11.47456965081045708204349353685, −10.88730876450574069570328766803, −9.435967208112401857955427064085, −8.576458012472630420451325309925, −7.87859041135076647221437368874, −6.68079948674430602220393414204, −4.77476252338135007775150815745, −3.94106124337465679784161016062, −3.26744404599477569037837159736,
1.48027246511287223912638691342, 3.19252179404765312929147401789, 4.77194773003216484007251457075, 6.50896988577796471147168127450, 7.02709456679633247494770507553, 7.79623929879750778191336394216, 8.966762091058463484567221700202, 10.37784994114388465514292477477, 11.37644089268480519317569779904, 11.95445565471089353745036965737