L(s) = 1 | + (0.707 − 0.707i)2-s + (−0.707 + 0.707i)3-s − 1.00i·4-s + (−1 + i)5-s + 1.00i·6-s − i·7-s + (−0.707 − 0.707i)8-s + 1.41i·10-s + (−0.707 − 0.707i)11-s + (0.707 + 0.707i)12-s + (0.707 − 0.707i)13-s + (−0.707 − 0.707i)14-s − 1.41i·15-s − 1.00·16-s − 1.41i·17-s + ⋯ |
L(s) = 1 | + (0.707 − 0.707i)2-s + (−0.707 + 0.707i)3-s − 1.00i·4-s + (−1 + i)5-s + 1.00i·6-s − i·7-s + (−0.707 − 0.707i)8-s + 1.41i·10-s + (−0.707 − 0.707i)11-s + (0.707 + 0.707i)12-s + (0.707 − 0.707i)13-s + (−0.707 − 0.707i)14-s − 1.41i·15-s − 1.00·16-s − 1.41i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7890592420\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7890592420\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 7 | \( 1 + iT \) |
| 13 | \( 1 + (-0.707 + 0.707i)T \) |
good | 3 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 5 | \( 1 + (1 - i)T - iT^{2} \) |
| 11 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 17 | \( 1 + 1.41iT - T^{2} \) |
| 19 | \( 1 + iT^{2} \) |
| 23 | \( 1 + iT - T^{2} \) |
| 29 | \( 1 + (-1 - i)T + iT^{2} \) |
| 31 | \( 1 - T + T^{2} \) |
| 37 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 41 | \( 1 + iT - T^{2} \) |
| 43 | \( 1 + (-1 + i)T - iT^{2} \) |
| 47 | \( 1 + T + T^{2} \) |
| 53 | \( 1 + (1 - i)T - iT^{2} \) |
| 59 | \( 1 - iT^{2} \) |
| 61 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 67 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 71 | \( 1 + 1.41T + T^{2} \) |
| 73 | \( 1 - iT - T^{2} \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T + T^{2} \) |
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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
−10.11716102445525895854563186831, −8.759257739843060350389750661832, −7.67297846177491838291006247880, −6.91605712789739376676277710285, −5.97360798460685568625389508265, −5.02267249056191296493971616614, −4.34120533308604372983357958369, −3.41591936516935319321740697790, −2.78591345017731068737579395427, −0.56508178756482463732034248234,
1.67919921483582931035253844426, 3.25991110029800798839618645724, 4.38245514607728526009341643846, 4.96623218448603570197381179414, 6.06736521703972960673813628034, 6.38188017801757557007102402365, 7.57346368040739108606761630599, 8.190881282423323150431888080681, 8.720924885347871844592637361848, 9.773633757690820479333975304650