L(s) = 1 | + (1.30 − 0.541i)2-s + (−1.13 + 1.30i)3-s + (1.41 − 1.41i)4-s + (−0.778 + 2.32i)6-s − 2.27i·7-s + (1.08 − 2.61i)8-s + (−0.414 − 2.97i)9-s − 4.20i·11-s + (0.239 + 3.45i)12-s + 3.21i·13-s + (−1.23 − 2.97i)14-s − 4i·16-s − 1.53i·17-s + (−2.14 − 3.65i)18-s + 4.82·19-s + ⋯ |
L(s) = 1 | + (0.923 − 0.382i)2-s + (−0.656 + 0.754i)3-s + (0.707 − 0.707i)4-s + (−0.317 + 0.948i)6-s − 0.859i·7-s + (0.382 − 0.923i)8-s + (−0.138 − 0.990i)9-s − 1.26i·11-s + (0.0692 + 0.997i)12-s + 0.891i·13-s + (−0.328 − 0.794i)14-s − i·16-s − 0.371i·17-s + (−0.506 − 0.862i)18-s + 1.10·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.445 + 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.445 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.72230 - 1.06645i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.72230 - 1.06645i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.30 + 0.541i)T \) |
| 3 | \( 1 + (1.13 - 1.30i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 2.27iT - 7T^{2} \) |
| 11 | \( 1 + 4.20iT - 11T^{2} \) |
| 13 | \( 1 - 3.21iT - 13T^{2} \) |
| 17 | \( 1 + 1.53iT - 17T^{2} \) |
| 19 | \( 1 - 4.82T + 19T^{2} \) |
| 23 | \( 1 + 1.08T + 23T^{2} \) |
| 29 | \( 1 - 1.74T + 29T^{2} \) |
| 31 | \( 1 + 6.82iT - 31T^{2} \) |
| 37 | \( 1 - 7.76iT - 37T^{2} \) |
| 41 | \( 1 - 2.46iT - 41T^{2} \) |
| 43 | \( 1 - 8.70T + 43T^{2} \) |
| 47 | \( 1 + 1.08T + 47T^{2} \) |
| 53 | \( 1 + 11.0T + 53T^{2} \) |
| 59 | \( 1 + 4.20iT - 59T^{2} \) |
| 61 | \( 1 - 8.48iT - 61T^{2} \) |
| 67 | \( 1 + 2.27T + 67T^{2} \) |
| 71 | \( 1 + 11.8T + 71T^{2} \) |
| 73 | \( 1 + 4.54T + 73T^{2} \) |
| 79 | \( 1 - 0.485iT - 79T^{2} \) |
| 83 | \( 1 - 6.94iT - 83T^{2} \) |
| 89 | \( 1 - 8.40iT - 89T^{2} \) |
| 97 | \( 1 - 10.9T + 97T^{2} \) |
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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
−10.71678132335841474566207268300, −9.934080207648075409232593064240, −9.131466042803462028972349671354, −7.60919516785693630193102903943, −6.52308690267103841640840255803, −5.80904597328183481249379998040, −4.76546189496142604088269181384, −3.96438560362147566244491759622, −3.03222557401893425980510585083, −0.971536533219536437717074305587,
1.85620693821595168788650718414, 3.00624950419357672622389322701, 4.59756692226858179581323359884, 5.44823797017475825603002401649, 6.09282541185030463839552094324, 7.21333811432865119035435399972, 7.71415195543690966921726022077, 8.837359448094761821388393109299, 10.20869740795998848056041088257, 11.09750467369372239290295323905