| L(s) = 1 | − 2-s + 4-s − 2.23i·5-s + (1.58 − 2.12i)7-s − 8-s + 2.23i·10-s + 1.41i·11-s + 3.16·13-s + (−1.58 + 2.12i)14-s + 16-s − 4.47i·17-s − 2.23i·20-s − 1.41i·22-s − 6·23-s − 5.00·25-s − 3.16·26-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.999i·5-s + (0.597 − 0.801i)7-s − 0.353·8-s + 0.707i·10-s + 0.426i·11-s + 0.877·13-s + (−0.422 + 0.566i)14-s + 0.250·16-s − 1.08i·17-s − 0.499i·20-s − 0.301i·22-s − 1.25·23-s − 1.00·25-s − 0.620·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0250 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0250 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.734716 - 0.753349i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.734716 - 0.753349i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 2.23iT \) |
| 7 | \( 1 + (-1.58 + 2.12i)T \) |
| good | 11 | \( 1 - 1.41iT - 11T^{2} \) |
| 13 | \( 1 - 3.16T + 13T^{2} \) |
| 17 | \( 1 + 4.47iT - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 + 2.82iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 4.24iT - 37T^{2} \) |
| 41 | \( 1 - 9.48T + 41T^{2} \) |
| 43 | \( 1 + 8.48iT - 43T^{2} \) |
| 47 | \( 1 + 4.47iT - 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 9.48T + 59T^{2} \) |
| 61 | \( 1 + 13.4iT - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 - 5.65iT - 71T^{2} \) |
| 73 | \( 1 + 6.32T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 8.94iT - 83T^{2} \) |
| 89 | \( 1 + 9.48T + 89T^{2} \) |
| 97 | \( 1 - 12.6T + 97T^{2} \) |
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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
−10.19019245128368008615632834623, −9.531236690328958957651902900364, −8.558511955016878264934298298333, −7.925117932707761620388852170211, −7.06977408034962942604364076625, −5.87469206114132527698504709593, −4.77625879851704816360809095817, −3.81808402610915940952594381156, −2.00033142574844190443925101690, −0.75552635560664395036088032976,
1.69990310217715932319361391860, 2.87145376950628508980656054161, 4.07690455696665579503348347992, 5.84316538225360915818631842009, 6.20168260108870078040702839289, 7.50703467499020019248506410240, 8.231401445990579585561071686273, 8.981022815354623021280794898161, 10.02241672483821613225587595146, 10.86960207532378637024513611208
