L(s) = 1 | + (−0.847 − 1.13i)2-s + (−0.242 + 1.71i)3-s + (−0.562 + 1.91i)4-s + (2.14 − 1.17i)6-s − 3.08i·7-s + (2.64 − 0.990i)8-s + (−2.88 − 0.831i)9-s + 2.54i·11-s + (−3.15 − 1.42i)12-s − 5.06i·13-s + (−3.49 + 2.61i)14-s + (−3.36 − 2.15i)16-s − 0.214i·17-s + (1.50 + 3.96i)18-s + 2.60·19-s + ⋯ |
L(s) = 1 | + (−0.599 − 0.800i)2-s + (−0.139 + 0.990i)3-s + (−0.281 + 0.959i)4-s + (0.876 − 0.481i)6-s − 1.16i·7-s + (0.936 − 0.350i)8-s + (−0.960 − 0.277i)9-s + 0.767i·11-s + (−0.910 − 0.412i)12-s − 1.40i·13-s + (−0.934 + 0.700i)14-s + (−0.841 − 0.539i)16-s − 0.0519i·17-s + (0.354 + 0.935i)18-s + 0.598·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.215 + 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.215 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.650169 - 0.522241i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.650169 - 0.522241i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.847 + 1.13i)T \) |
| 3 | \( 1 + (0.242 - 1.71i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 3.08iT - 7T^{2} \) |
| 11 | \( 1 - 2.54iT - 11T^{2} \) |
| 13 | \( 1 + 5.06iT - 13T^{2} \) |
| 17 | \( 1 + 0.214iT - 17T^{2} \) |
| 19 | \( 1 - 2.60T + 19T^{2} \) |
| 23 | \( 1 + 4.47T + 23T^{2} \) |
| 29 | \( 1 - 7.86T + 29T^{2} \) |
| 31 | \( 1 + 4.58iT - 31T^{2} \) |
| 37 | \( 1 + 7.67iT - 37T^{2} \) |
| 41 | \( 1 + 9.26iT - 41T^{2} \) |
| 43 | \( 1 - 11.4T + 43T^{2} \) |
| 47 | \( 1 - 10.5T + 47T^{2} \) |
| 53 | \( 1 + 9.51T + 53T^{2} \) |
| 59 | \( 1 + 0.428iT - 59T^{2} \) |
| 61 | \( 1 + 1.11iT - 61T^{2} \) |
| 67 | \( 1 - 2.35T + 67T^{2} \) |
| 71 | \( 1 + 6.12T + 71T^{2} \) |
| 73 | \( 1 + 12.0T + 73T^{2} \) |
| 79 | \( 1 - 11.6iT - 79T^{2} \) |
| 83 | \( 1 + 2.29iT - 83T^{2} \) |
| 89 | \( 1 + 12.4iT - 89T^{2} \) |
| 97 | \( 1 + 8.04T + 97T^{2} \) |
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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
−10.42996747864276660317448295448, −9.903809240965055195231873225996, −9.031794345575070776480179482444, −7.932551399209736289352525501108, −7.28523633909897291279756027883, −5.70629300620574308749879433697, −4.45807803162286936409158096665, −3.79692131139286634267094207330, −2.62546602314147266707028848444, −0.63636678375477004273418615358,
1.36847570390868223969752584787, 2.67502191356448187251184923893, 4.69351408840131198153494219848, 5.86852254021906752135009319344, 6.31004249524442098535499977868, 7.27890540088838916915010904131, 8.304788113537817829759166962098, 8.787454954043408484830673862587, 9.660095213139341331603201207841, 10.90740717981280069660375176010