L(s) = 1 | + (2.12 − 2.12i)2-s − 0.999i·4-s + (−9 − 9i)7-s + (14.8 + 14.8i)8-s − 38.1i·11-s + (63 − 63i)13-s − 38.1·14-s + 71·16-s + (29.6 − 29.6i)17-s − 70i·19-s + (−81 − 81i)22-s + (72.1 + 72.1i)23-s − 267. i·26-s + (−8.99 + 8.99i)28-s − 229.·29-s + ⋯ |
L(s) = 1 | + (0.749 − 0.749i)2-s − 0.124i·4-s + (−0.485 − 0.485i)7-s + (0.656 + 0.656i)8-s − 1.04i·11-s + (1.34 − 1.34i)13-s − 0.728·14-s + 1.10·16-s + (0.423 − 0.423i)17-s − 0.845i·19-s + (−0.784 − 0.784i)22-s + (0.653 + 0.653i)23-s − 2.01i·26-s + (−0.0607 + 0.0607i)28-s − 1.46·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0618 + 0.998i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0618 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.93057 - 1.81461i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.93057 - 1.81461i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (-2.12 + 2.12i)T - 8iT^{2} \) |
| 7 | \( 1 + (9 + 9i)T + 343iT^{2} \) |
| 11 | \( 1 + 38.1iT - 1.33e3T^{2} \) |
| 13 | \( 1 + (-63 + 63i)T - 2.19e3iT^{2} \) |
| 17 | \( 1 + (-29.6 + 29.6i)T - 4.91e3iT^{2} \) |
| 19 | \( 1 + 70iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (-72.1 - 72.1i)T + 1.21e4iT^{2} \) |
| 29 | \( 1 + 229.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 196T + 2.97e4T^{2} \) |
| 37 | \( 1 + (-207 - 207i)T + 5.06e4iT^{2} \) |
| 41 | \( 1 + 267. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + (144 - 144i)T - 7.95e4iT^{2} \) |
| 47 | \( 1 + (356. - 356. i)T - 1.03e5iT^{2} \) |
| 53 | \( 1 + (224. + 224. i)T + 1.48e5iT^{2} \) |
| 59 | \( 1 + 267.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 322T + 2.26e5T^{2} \) |
| 67 | \( 1 + (-378 - 378i)T + 3.00e5iT^{2} \) |
| 71 | \( 1 - 840. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-378 + 378i)T - 3.89e5iT^{2} \) |
| 79 | \( 1 + 488iT - 4.93e5T^{2} \) |
| 83 | \( 1 + (-772. - 772. i)T + 5.71e5iT^{2} \) |
| 89 | \( 1 + 267.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (252 + 252i)T + 9.12e5iT^{2} \) |
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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
−11.35141726013058466014390186558, −11.06150234015556429580939962494, −9.904997173239584814307937341986, −8.545552111989571715238918445272, −7.63988272991155122013862636542, −6.16614641491581033910791772439, −5.07546082215536363680178892796, −3.59585113682796791094767826017, −3.01883206035402986110805243031, −0.967239817515722642330527394836,
1.66612519306822245890503471034, 3.69765596098251923377977862788, 4.71668649373565350132669350449, 6.02759790741328477042840901861, 6.58995680535944119871169699128, 7.78457504866236935251638874729, 9.115998302257311390789916180716, 9.991369552569039143728341578901, 11.14882423179248492001429278758, 12.34318794769150897373086856561