L(s) = 1 | + (−0.309 + 0.951i)4-s + (0.809 + 0.587i)5-s + (−0.809 + 0.587i)9-s + (−0.809 − 0.587i)16-s + (−0.809 + 0.587i)20-s + (0.309 + 0.951i)25-s + (1.61 − 1.17i)31-s + (−0.309 − 0.951i)36-s − 45-s + (0.809 + 0.587i)49-s + (0.618 − 1.90i)59-s + (0.809 − 0.587i)64-s + (−1.61 − 1.17i)71-s + (−0.309 − 0.951i)80-s + (0.309 − 0.951i)81-s + ⋯ |
L(s) = 1 | + (−0.309 + 0.951i)4-s + (0.809 + 0.587i)5-s + (−0.809 + 0.587i)9-s + (−0.809 − 0.587i)16-s + (−0.809 + 0.587i)20-s + (0.309 + 0.951i)25-s + (1.61 − 1.17i)31-s + (−0.309 − 0.951i)36-s − 45-s + (0.809 + 0.587i)49-s + (0.618 − 1.90i)59-s + (0.809 − 0.587i)64-s + (−1.61 − 1.17i)71-s + (−0.309 − 0.951i)80-s + (0.309 − 0.951i)81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.331 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.331 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9095748749\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9095748749\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-0.809 - 0.587i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 3 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 7 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 13 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 17 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 19 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 31 | \( 1 + (-1.61 + 1.17i)T + (0.309 - 0.951i)T^{2} \) |
| 37 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 41 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 53 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 59 | \( 1 + (-0.618 + 1.90i)T + (-0.809 - 0.587i)T^{2} \) |
| 61 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (1.61 + 1.17i)T + (0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 79 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 83 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 89 | \( 1 + 2T + T^{2} \) |
| 97 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
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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
−11.10640127018173265016393251240, −10.12964911275597204366139209982, −9.267529649550375104435091071579, −8.369619885117606154444801008001, −7.61158637165459356203835673742, −6.56239259359205143526452280480, −5.60486442864822944737407579787, −4.48621505072117923466162257646, −3.16176684644452720127222228134, −2.32338853559295554678607845651,
1.18772841557469463712064700282, 2.67512171025571984692626748087, 4.32080135321533807294330681642, 5.33922675539379233656769861538, 5.97662165303622354065774921055, 6.84580056707665313575172464053, 8.488666069473609464582474879231, 8.932122898846750616824873044124, 9.879803649720647772225496719705, 10.42924923232818593886649707271