L(s) = 1 | + (3.34 − 2.18i)2-s + 5.19i·3-s + (6.43 − 14.6i)4-s + (11.3 + 17.4i)6-s − 1.39i·7-s + (−10.5 − 63.1i)8-s − 27·9-s + 164. i·11-s + (76.1 + 33.4i)12-s + 158.·13-s + (−3.05 − 4.67i)14-s + (−173. − 188. i)16-s + 548.·17-s + (−90.4 + 59.0i)18-s + 25.6i·19-s + ⋯ |
L(s) = 1 | + (0.837 − 0.546i)2-s + 0.577i·3-s + (0.401 − 0.915i)4-s + (0.315 + 0.483i)6-s − 0.0284i·7-s + (−0.164 − 0.986i)8-s − 0.333·9-s + 1.36i·11-s + (0.528 + 0.232i)12-s + 0.938·13-s + (−0.0155 − 0.0238i)14-s + (−0.676 − 0.736i)16-s + 1.89·17-s + (−0.279 + 0.182i)18-s + 0.0711i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.915 + 0.401i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.915 + 0.401i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(3.546565430\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.546565430\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-3.34 + 2.18i)T \) |
| 3 | \( 1 - 5.19iT \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 1.39iT - 2.40e3T^{2} \) |
| 11 | \( 1 - 164. iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 158.T + 2.85e4T^{2} \) |
| 17 | \( 1 - 548.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 25.6iT - 1.30e5T^{2} \) |
| 23 | \( 1 + 730. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 773.T + 7.07e5T^{2} \) |
| 31 | \( 1 - 194. iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 1.37e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + 647.T + 2.82e6T^{2} \) |
| 43 | \( 1 - 1.50e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 647. iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 4.40e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + 5.97e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 560.T + 1.38e7T^{2} \) |
| 67 | \( 1 + 2.98e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 2.67e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 2.64e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 3.15e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 - 1.30e4iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 7.11e3T + 6.27e7T^{2} \) |
| 97 | \( 1 + 5.31e3T + 8.85e7T^{2} \) |
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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
−11.03170518752591758231634258114, −10.17873530608519783686242630708, −9.620735801087909697662129120305, −8.223589226553319146683596973433, −6.87639433648814828264861116102, −5.78741914356767626730609234793, −4.75454423195855292733306033861, −3.84451331128542449359083331965, −2.64477781017275777970719183618, −1.11938681543897745432313218844,
1.11219033850203479403140953174, 2.94229018099007356172275947034, 3.81251339212111181320367205512, 5.55537515935006795797737073084, 5.94565223803027208199687426704, 7.22083872435305188349440649145, 8.070598020964429243994715362751, 8.869175232898134194285310888499, 10.45812575166350755271129090742, 11.59096103404660229263803544732