L(s) = 1 | + (−0.923 + 0.382i)2-s + (0.707 + 0.707i)3-s + (0.707 − 0.707i)4-s + (1.30 − 1.30i)5-s + (−0.923 − 0.382i)6-s + (−0.382 + 0.923i)8-s + 1.00i·9-s + (−0.707 + 1.70i)10-s + (−0.541 + 0.541i)11-s + 12-s + (−0.707 − 0.707i)13-s + 1.84·15-s − i·16-s + (−0.382 − 0.923i)18-s − 1.84i·20-s + ⋯ |
L(s) = 1 | + (−0.923 + 0.382i)2-s + (0.707 + 0.707i)3-s + (0.707 − 0.707i)4-s + (1.30 − 1.30i)5-s + (−0.923 − 0.382i)6-s + (−0.382 + 0.923i)8-s + 1.00i·9-s + (−0.707 + 1.70i)10-s + (−0.541 + 0.541i)11-s + 12-s + (−0.707 − 0.707i)13-s + 1.84·15-s − i·16-s + (−0.382 − 0.923i)18-s − 1.84i·20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8961996099\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8961996099\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.923 - 0.382i)T \) |
| 3 | \( 1 + (-0.707 - 0.707i)T \) |
| 13 | \( 1 + (0.707 + 0.707i)T \) |
good | 5 | \( 1 + (-1.30 + 1.30i)T - iT^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 + (0.541 - 0.541i)T - iT^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - iT^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 - iT^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 - 1.84iT - T^{2} \) |
| 43 | \( 1 + (1 - i)T - iT^{2} \) |
| 47 | \( 1 + 0.765T + T^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 + (-1.30 + 1.30i)T - iT^{2} \) |
| 61 | \( 1 + iT^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 - 0.765iT - T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 - 1.41T + T^{2} \) |
| 83 | \( 1 + (1.30 + 1.30i)T + iT^{2} \) |
| 89 | \( 1 + 0.765iT - T^{2} \) |
| 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
−10.16251444993228925113357358920, −9.913169754619372803195459801714, −9.266865605334552413384404666163, −8.355874918583961019319674442757, −7.81497252812353444474036567915, −6.39280428030400885553877182178, −5.22209901798347140618873888210, −4.83691777022271644997221636205, −2.74691665918881862604545528709, −1.68807888608735225006993147654,
1.87201397448150074188789799761, 2.56558107265464721939571014085, 3.49626980091899211900167495765, 5.70882220329959930277198826926, 6.76133677002433702921005084271, 7.12937920842458185781815718334, 8.228766360164405810188773788314, 9.166082201749211922984570747704, 9.849293193517854592864350317746, 10.53922636886570934371481576867