L(s) = 1 | + (1.05 + 1.82i)3-s + i·5-s + (2.22 + 1.28i)7-s + (−0.717 + 1.24i)9-s + (−0.677 + 0.391i)11-s + (−3.60 + 0.00176i)13-s + (−1.82 + 1.05i)15-s + (−2.71 + 4.70i)17-s + (2.95 + 1.70i)19-s + 5.40i·21-s + (1.35 + 2.33i)23-s − 25-s + 3.29·27-s + (1.43 + 2.48i)29-s − 4.66i·31-s + ⋯ |
L(s) = 1 | + (0.607 + 1.05i)3-s + 0.447i·5-s + (0.840 + 0.485i)7-s + (−0.239 + 0.414i)9-s + (−0.204 + 0.117i)11-s + (−0.999 + 0.000490i)13-s + (−0.470 + 0.271i)15-s + (−0.658 + 1.14i)17-s + (0.677 + 0.391i)19-s + 1.18i·21-s + (0.281 + 0.487i)23-s − 0.200·25-s + 0.634·27-s + (0.266 + 0.461i)29-s − 0.838i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.489 - 0.872i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.489 - 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.941854719\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.941854719\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 13 | \( 1 + (3.60 - 0.00176i)T \) |
good | 3 | \( 1 + (-1.05 - 1.82i)T + (-1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 + (-2.22 - 1.28i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (0.677 - 0.391i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (2.71 - 4.70i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.95 - 1.70i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.35 - 2.33i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.43 - 2.48i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 4.66iT - 31T^{2} \) |
| 37 | \( 1 + (4.67 - 2.69i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.54 + 2.04i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.06 + 3.57i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 1.17iT - 47T^{2} \) |
| 53 | \( 1 + 7.43T + 53T^{2} \) |
| 59 | \( 1 + (-11.6 - 6.72i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.200 + 0.347i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (11.8 - 6.84i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (4.07 + 2.35i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 7.33iT - 73T^{2} \) |
| 79 | \( 1 - 8.10T + 79T^{2} \) |
| 83 | \( 1 - 14.2iT - 83T^{2} \) |
| 89 | \( 1 + (-11.5 + 6.65i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.98 - 1.72i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.17571588972260413166456274697, −9.398384069949609215883440941228, −8.660649184609190072271383613396, −7.88115755135265426024653488792, −6.96857058619894779086840944381, −5.72088861936471450061607013271, −4.85825987639616060920795358644, −4.01714757697259216639515425827, −3.00667656023757436189732133848, −1.93586101637184683699111916361,
0.825371299803603391437084138966, 2.04595486388894785260706922479, 2.96887866413347988828853327430, 4.59559370635780452279567935035, 5.08999558896499106058631291770, 6.56319521015045803126609494664, 7.38969174163250297341161614981, 7.79278610374509520338138719122, 8.703671338118574388376051217999, 9.438047054268041331902757463613