L(s) = 1 | + (−0.222 + 0.974i)2-s + (−0.900 − 0.433i)4-s + (−0.277 + 1.21i)5-s + (0.623 − 0.781i)8-s + (0.623 − 0.781i)9-s + (−1.12 − 0.541i)10-s + (−1.12 − 1.40i)13-s + (0.623 + 0.781i)16-s − 0.445·17-s + (0.623 + 0.781i)18-s + (0.777 − 0.974i)20-s + (−0.499 − 0.240i)25-s + (1.62 − 0.781i)26-s + (0.623 + 0.781i)29-s + (−0.900 + 0.433i)32-s + ⋯ |
L(s) = 1 | + (−0.222 + 0.974i)2-s + (−0.900 − 0.433i)4-s + (−0.277 + 1.21i)5-s + (0.623 − 0.781i)8-s + (0.623 − 0.781i)9-s + (−1.12 − 0.541i)10-s + (−1.12 − 1.40i)13-s + (0.623 + 0.781i)16-s − 0.445·17-s + (0.623 + 0.781i)18-s + (0.777 − 0.974i)20-s + (−0.499 − 0.240i)25-s + (1.62 − 0.781i)26-s + (0.623 + 0.781i)29-s + (−0.900 + 0.433i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.218 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.218 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4923534836\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4923534836\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.222 - 0.974i)T \) |
| 29 | \( 1 + (-0.623 - 0.781i)T \) |
good | 3 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 5 | \( 1 + (0.277 - 1.21i)T + (-0.900 - 0.433i)T^{2} \) |
| 7 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 11 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 13 | \( 1 + (1.12 + 1.40i)T + (-0.222 + 0.974i)T^{2} \) |
| 17 | \( 1 + 0.445T + T^{2} \) |
| 19 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 23 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 31 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 37 | \( 1 + (1.12 - 1.40i)T + (-0.222 - 0.974i)T^{2} \) |
| 41 | \( 1 + 1.80T + T^{2} \) |
| 43 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 47 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 53 | \( 1 + (-0.0990 + 0.433i)T + (-0.900 - 0.433i)T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + (-0.400 + 0.193i)T + (0.623 - 0.781i)T^{2} \) |
| 67 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 71 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 73 | \( 1 + (-0.0990 - 0.433i)T + (-0.900 + 0.433i)T^{2} \) |
| 79 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 83 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 89 | \( 1 + (0.277 - 1.21i)T + (-0.900 - 0.433i)T^{2} \) |
| 97 | \( 1 + (-1.62 - 0.781i)T + (0.623 + 0.781i)T^{2} \) |
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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
−14.40106795523275260690839766768, −13.20258346073359923438809195216, −12.08940785886430973806973782692, −10.48640617364733001908786851926, −9.915717125772257711920818893238, −8.441326966483433793615787218850, −7.22375586503258266377073054359, −6.61775284997599411631168994524, −5.07114890143137279357542216185, −3.36832508695949639784764112068,
1.98953106651989121279289648790, 4.25834907118962172763883245254, 4.98198964292263886909688124692, 7.30024813219237327687015242778, 8.527768779861998348410218713973, 9.381670556886473611805034923313, 10.42584065434807916780992551585, 11.72552978036225993307987305579, 12.37500403247397481488490896898, 13.32241310806183552707423500066