L(s) = 1 | + (0.707 − 0.707i)2-s + (1.60 + 0.639i)3-s − 1.00i·4-s + (1.63 − 1.52i)5-s + (1.59 − 0.685i)6-s + (0.772 + 0.772i)7-s + (−0.707 − 0.707i)8-s + (2.18 + 2.05i)9-s + (0.0731 − 2.23i)10-s + 1.36i·11-s + (0.639 − 1.60i)12-s + (0.808 − 0.808i)13-s + 1.09·14-s + (3.60 − 1.41i)15-s − 1.00·16-s + (−2.66 + 2.66i)17-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s + (0.929 + 0.369i)3-s − 0.500i·4-s + (0.729 − 0.683i)5-s + (0.649 − 0.279i)6-s + (0.291 + 0.291i)7-s + (−0.250 − 0.250i)8-s + (0.727 + 0.686i)9-s + (0.0231 − 0.706i)10-s + 0.411i·11-s + (0.184 − 0.464i)12-s + (0.224 − 0.224i)13-s + 0.291·14-s + (0.930 − 0.365i)15-s − 0.250·16-s + (−0.645 + 0.645i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.800 + 0.599i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.800 + 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.85585 - 0.950839i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.85585 - 0.950839i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 3 | \( 1 + (-1.60 - 0.639i)T \) |
| 5 | \( 1 + (-1.63 + 1.52i)T \) |
| 23 | \( 1 + (0.707 + 0.707i)T \) |
good | 7 | \( 1 + (-0.772 - 0.772i)T + 7iT^{2} \) |
| 11 | \( 1 - 1.36iT - 11T^{2} \) |
| 13 | \( 1 + (-0.808 + 0.808i)T - 13iT^{2} \) |
| 17 | \( 1 + (2.66 - 2.66i)T - 17iT^{2} \) |
| 19 | \( 1 + 1.89iT - 19T^{2} \) |
| 29 | \( 1 - 1.81T + 29T^{2} \) |
| 31 | \( 1 + 5.22T + 31T^{2} \) |
| 37 | \( 1 + (7.05 + 7.05i)T + 37iT^{2} \) |
| 41 | \( 1 - 7.21iT - 41T^{2} \) |
| 43 | \( 1 + (-4.99 + 4.99i)T - 43iT^{2} \) |
| 47 | \( 1 + (7.28 - 7.28i)T - 47iT^{2} \) |
| 53 | \( 1 + (-1.52 - 1.52i)T + 53iT^{2} \) |
| 59 | \( 1 - 5.33T + 59T^{2} \) |
| 61 | \( 1 + 12.6T + 61T^{2} \) |
| 67 | \( 1 + (-8.07 - 8.07i)T + 67iT^{2} \) |
| 71 | \( 1 - 2.85iT - 71T^{2} \) |
| 73 | \( 1 + (-8.07 + 8.07i)T - 73iT^{2} \) |
| 79 | \( 1 - 4.48iT - 79T^{2} \) |
| 83 | \( 1 + (-0.994 - 0.994i)T + 83iT^{2} \) |
| 89 | \( 1 + 3.62T + 89T^{2} \) |
| 97 | \( 1 + (0.314 + 0.314i)T + 97iT^{2} \) |
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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
−10.36137594243894973940380785056, −9.461801134786851796241914622458, −8.881168381516293399572993594684, −8.056804756645744802501405361693, −6.76755638404419792022342131298, −5.56325621245298519115890850906, −4.74193144217355779758213534108, −3.84473125828153591183917066693, −2.51059087696502408265556749226, −1.65046029184048056496399012737,
1.80338837680355960816918943956, 2.96344296792773421364226033266, 3.88965703031307836127482189656, 5.17600984917955499864302510260, 6.33929991098162151058444581664, 6.96474950432231704725264594884, 7.80984736122436013763045521482, 8.737953130255844835256699416486, 9.514024946491207621209258827585, 10.50774120633922082419443223273