L(s) = 1 | + (0.623 + 0.781i)3-s + (−0.222 − 0.974i)4-s − 1.80·7-s + (−0.222 + 0.974i)9-s + (0.623 − 0.781i)12-s + (0.400 + 0.193i)13-s + (−0.900 + 0.433i)16-s + (−0.277 − 1.21i)19-s + (−1.12 − 1.40i)21-s + (0.623 + 0.781i)25-s + (−0.900 + 0.433i)27-s + (0.400 + 1.75i)28-s + (0.777 − 0.974i)31-s + 36-s − 0.445·37-s + ⋯ |
L(s) = 1 | + (0.623 + 0.781i)3-s + (−0.222 − 0.974i)4-s − 1.80·7-s + (−0.222 + 0.974i)9-s + (0.623 − 0.781i)12-s + (0.400 + 0.193i)13-s + (−0.900 + 0.433i)16-s + (−0.277 − 1.21i)19-s + (−1.12 − 1.40i)21-s + (0.623 + 0.781i)25-s + (−0.900 + 0.433i)27-s + (0.400 + 1.75i)28-s + (0.777 − 0.974i)31-s + 36-s − 0.445·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6200503732\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6200503732\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.623 - 0.781i)T \) |
| 43 | \( 1 + (-0.623 - 0.781i)T \) |
good | 2 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 5 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 7 | \( 1 + 1.80T + T^{2} \) |
| 11 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 13 | \( 1 + (-0.400 - 0.193i)T + (0.623 + 0.781i)T^{2} \) |
| 17 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 19 | \( 1 + (0.277 + 1.21i)T + (-0.900 + 0.433i)T^{2} \) |
| 23 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 29 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 31 | \( 1 + (-0.777 + 0.974i)T + (-0.222 - 0.974i)T^{2} \) |
| 37 | \( 1 + 0.445T + T^{2} \) |
| 41 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 47 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 53 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 59 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 61 | \( 1 + (0.277 + 0.347i)T + (-0.222 + 0.974i)T^{2} \) |
| 67 | \( 1 + (-0.400 - 1.75i)T + (-0.900 + 0.433i)T^{2} \) |
| 71 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 73 | \( 1 + (1.12 + 0.541i)T + (0.623 + 0.781i)T^{2} \) |
| 79 | \( 1 + 1.80T + T^{2} \) |
| 83 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 89 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 97 | \( 1 + (-0.400 + 1.75i)T + (-0.900 - 0.433i)T^{2} \) |
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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
−13.55402544841751887749647923387, −13.02902074927801376082986676113, −11.23085784124744062990294406030, −10.23070536882503965117717150118, −9.486667543166512574391777251332, −8.810426837993992849677089173916, −6.91966399279496071088329594102, −5.78240941938146748824322621911, −4.33280687056394933560473687743, −2.88325771425224940394628159345,
2.86012454576471401070129813826, 3.78897603568362382235659278873, 6.20105180911059626308592776543, 7.07768067308750310483855913972, 8.277214629916687585408263023458, 9.115486154676504315771557643676, 10.26264333481996414389821421833, 12.12023276593579841919249826409, 12.57488235395403764659415280905, 13.36500631173665182626393085077