L(s) = 1 | + (0.707 + 0.707i)2-s + (−0.292 − 1.70i)3-s + 1.00i·4-s + 2.23i·5-s + (0.999 − 1.41i)6-s + (−0.707 + 0.707i)8-s + (−2.82 + i)9-s + (−1.58 + 1.58i)10-s − 2.82i·11-s + (1.70 − 0.292i)12-s + (4.16 + 4.16i)13-s + (3.81 − 0.654i)15-s − 1.00·16-s + (3.65 + 3.65i)17-s + (−2.70 − 1.29i)18-s + 5.16i·19-s + ⋯ |
L(s) = 1 | + (0.499 + 0.499i)2-s + (−0.169 − 0.985i)3-s + 0.500i·4-s + 0.999i·5-s + (0.408 − 0.577i)6-s + (−0.250 + 0.250i)8-s + (−0.942 + 0.333i)9-s + (−0.500 + 0.500i)10-s − 0.852i·11-s + (0.492 − 0.0845i)12-s + (1.15 + 1.15i)13-s + (0.985 − 0.169i)15-s − 0.250·16-s + (0.885 + 0.885i)17-s + (−0.638 − 0.304i)18-s + 1.18i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.374 - 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.374 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.46904 + 0.991223i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.46904 + 0.991223i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.292 + 1.70i)T \) |
| 5 | \( 1 - 2.23iT \) |
| 23 | \( 1 + (0.707 - 0.707i)T \) |
good | 7 | \( 1 - 7iT^{2} \) |
| 11 | \( 1 + 2.82iT - 11T^{2} \) |
| 13 | \( 1 + (-4.16 - 4.16i)T + 13iT^{2} \) |
| 17 | \( 1 + (-3.65 - 3.65i)T + 17iT^{2} \) |
| 19 | \( 1 - 5.16iT - 19T^{2} \) |
| 29 | \( 1 + 4.47T + 29T^{2} \) |
| 31 | \( 1 - 10.3T + 31T^{2} \) |
| 37 | \( 1 + (-5.16 + 5.16i)T - 37iT^{2} \) |
| 41 | \( 1 - 6.11iT - 41T^{2} \) |
| 43 | \( 1 + (2.83 + 2.83i)T + 43iT^{2} \) |
| 47 | \( 1 + (8.71 + 8.71i)T + 47iT^{2} \) |
| 53 | \( 1 + (6.70 - 6.70i)T - 53iT^{2} \) |
| 59 | \( 1 - 9.89T + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 + (-5.16 + 5.16i)T - 67iT^{2} \) |
| 71 | \( 1 + 3.05iT - 71T^{2} \) |
| 73 | \( 1 + (7.32 + 7.32i)T + 73iT^{2} \) |
| 79 | \( 1 - 3.16iT - 79T^{2} \) |
| 83 | \( 1 + (5.88 - 5.88i)T - 83iT^{2} \) |
| 89 | \( 1 + 0.458T + 89T^{2} \) |
| 97 | \( 1 + (-1.16 + 1.16i)T - 97iT^{2} \) |
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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
−10.98051993249307655722818489301, −9.810392774052877507446289001986, −8.394024185977614352585341213070, −7.988251071895501440387636947324, −6.89343413772146271090985666845, −6.15683189908640842345303043588, −5.76422129720295689159940879604, −3.98192286443537433050914471755, −3.10980569070032016261842038759, −1.65352922810162116145508389694,
0.884448797860282078120196194643, 2.77402710710490542156853599727, 3.85035966592552600181741963429, 4.84952980948401669702319239846, 5.33153597054205838126672408656, 6.41500998671630582561970026091, 7.959438700899699733471722993977, 8.771865439911610807124381345785, 9.782198939356716686589244349355, 10.10761456781451183238510673478