L(s) = 1 | + (1.29 − 0.569i)2-s + (0.908 − 1.47i)3-s + (1.35 − 1.47i)4-s + (0.335 − 2.42i)6-s + 2.50i·7-s + (0.908 − 2.67i)8-s + (−1.35 − 2.67i)9-s − 3.36·11-s + (−0.948 − 3.33i)12-s + 3.70·13-s + (1.42 + 3.24i)14-s + (−0.350 − 3.98i)16-s + 7.63i·17-s + (−3.27 − 2.69i)18-s + 0.440i·19-s + ⋯ |
L(s) = 1 | + (0.915 − 0.402i)2-s + (0.524 − 0.851i)3-s + (0.675 − 0.737i)4-s + (0.136 − 0.990i)6-s + 0.948i·7-s + (0.321 − 0.947i)8-s + (−0.450 − 0.892i)9-s − 1.01·11-s + (−0.273 − 0.961i)12-s + 1.02·13-s + (0.382 + 0.868i)14-s + (−0.0876 − 0.996i)16-s + 1.85i·17-s + (−0.771 − 0.635i)18-s + 0.100i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.273 + 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.273 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.94235 - 1.46644i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.94235 - 1.46644i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.29 + 0.569i)T \) |
| 3 | \( 1 + (-0.908 + 1.47i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 2.50iT - 7T^{2} \) |
| 11 | \( 1 + 3.36T + 11T^{2} \) |
| 13 | \( 1 - 3.70T + 13T^{2} \) |
| 17 | \( 1 - 7.63iT - 17T^{2} \) |
| 19 | \( 1 - 0.440iT - 19T^{2} \) |
| 23 | \( 1 + 5.17T + 23T^{2} \) |
| 29 | \( 1 + 2.27iT - 29T^{2} \) |
| 31 | \( 1 + 3.39iT - 31T^{2} \) |
| 37 | \( 1 - 7.40T + 37T^{2} \) |
| 41 | \( 1 - 3.07iT - 41T^{2} \) |
| 43 | \( 1 - 8.40iT - 43T^{2} \) |
| 47 | \( 1 - 3.63T + 47T^{2} \) |
| 53 | \( 1 + 2.27iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 5.70T + 61T^{2} \) |
| 67 | \( 1 + 5.45iT - 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 + 1.29T + 73T^{2} \) |
| 79 | \( 1 - 5.01iT - 79T^{2} \) |
| 83 | \( 1 - 1.81T + 83T^{2} \) |
| 89 | \( 1 + 5.35iT - 89T^{2} \) |
| 97 | \( 1 + 11.1T + 97T^{2} \) |
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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
−11.80193649059811110940469741931, −10.90202690298535155822783544485, −9.793174117311338645501834698234, −8.500939289098503289618093821104, −7.75221569498017239048711493278, −6.12713087926160531535447748669, −5.90092286689973119254377468829, −4.10958492030487539044182663372, −2.85289373973446634562174677562, −1.75944430066183494829233942261,
2.66126834058225395264992515093, 3.76368590506245872423999576078, 4.71987970962092421022380363894, 5.69995619080790655377077316888, 7.16487443460115922889168030054, 7.916997665243586305881253458680, 9.000174189738424919452102987717, 10.30501254685784823537731051954, 10.93202194136890095290553173597, 11.93891250027350734275920577953