L(s) = 1 | + (−0.707 − 0.707i)7-s + (0.707 − 0.707i)9-s + (−0.707 − 0.292i)11-s + (1 − i)23-s + (−0.707 − 0.707i)25-s + (−0.707 + 0.292i)29-s + (0.707 − 1.70i)37-s + (−1.70 − 0.707i)43-s + 1.00i·49-s + (1.70 + 0.707i)53-s − 1.00·63-s + (1.70 − 0.707i)67-s + (0.292 + 0.707i)77-s + 1.41i·79-s − 1.00i·81-s + ⋯ |
L(s) = 1 | + (−0.707 − 0.707i)7-s + (0.707 − 0.707i)9-s + (−0.707 − 0.292i)11-s + (1 − i)23-s + (−0.707 − 0.707i)25-s + (−0.707 + 0.292i)29-s + (0.707 − 1.70i)37-s + (−1.70 − 0.707i)43-s + 1.00i·49-s + (1.70 + 0.707i)53-s − 1.00·63-s + (1.70 − 0.707i)67-s + (0.292 + 0.707i)77-s + 1.41i·79-s − 1.00i·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.195 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.195 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9515569652\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9515569652\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (0.707 + 0.707i)T \) |
good | 3 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 5 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 11 | \( 1 + (0.707 + 0.292i)T + (0.707 + 0.707i)T^{2} \) |
| 13 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 23 | \( 1 + (-1 + i)T - iT^{2} \) |
| 29 | \( 1 + (0.707 - 0.292i)T + (0.707 - 0.707i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (-0.707 + 1.70i)T + (-0.707 - 0.707i)T^{2} \) |
| 41 | \( 1 + iT^{2} \) |
| 43 | \( 1 + (1.70 + 0.707i)T + (0.707 + 0.707i)T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + (-1.70 - 0.707i)T + (0.707 + 0.707i)T^{2} \) |
| 59 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 61 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 67 | \( 1 + (-1.70 + 0.707i)T + (0.707 - 0.707i)T^{2} \) |
| 71 | \( 1 + iT^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 - 1.41iT - T^{2} \) |
| 83 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 89 | \( 1 - iT^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.369956740985786274677488992391, −8.581508575208752005405685829326, −7.57583599878106890605847855900, −6.94275584401190643759361144731, −6.23276962458764072875551943722, −5.24304145565766964589301061534, −4.16009018173881436333901245907, −3.50345252333540068860612328111, −2.35015299267904367884262980792, −0.72326751539914974162141785262,
1.70070567771751940877389402323, 2.75221736580233919425873058706, 3.71071883275013071287590434628, 4.93392771874520524762208233616, 5.46685489050827273220398100456, 6.51609717971453323263356688353, 7.31356447045284481528231322184, 8.010267609747506675184893420611, 8.897396295218549634700505891977, 9.878295106049360111944877052360