L(s) = 1 | − 3.20i·2-s − 29.5i·3-s + 21.7·4-s − 94.8·6-s + 11.5i·7-s − 172. i·8-s − 632.·9-s − 596.·11-s − 642. i·12-s + 169i·13-s + 37.1·14-s + 142.·16-s − 2.09e3i·17-s + 2.02e3i·18-s − 35.4·19-s + ⋯ |
L(s) = 1 | − 0.566i·2-s − 1.89i·3-s + 0.678·4-s − 1.07·6-s + 0.0892i·7-s − 0.951i·8-s − 2.60·9-s − 1.48·11-s − 1.28i·12-s + 0.277i·13-s + 0.0506·14-s + 0.139·16-s − 1.75i·17-s + 1.47i·18-s − 0.0225·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.7659285357\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7659285357\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 13 | \( 1 - 169iT \) |
good | 2 | \( 1 + 3.20iT - 32T^{2} \) |
| 3 | \( 1 + 29.5iT - 243T^{2} \) |
| 7 | \( 1 - 11.5iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 596.T + 1.61e5T^{2} \) |
| 17 | \( 1 + 2.09e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 35.4T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.78e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 370.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.05e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 4.12e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 1.81e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 7.90e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 1.31e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 3.82e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 1.78e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 7.39e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.33e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 3.31e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.08e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 1.77e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 8.42e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 4.64e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.54e5iT - 8.58e9T^{2} \) |
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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
−10.26464985074492341994357727996, −8.931060139785972302820022867597, −7.59707640601827919319344262269, −7.36955737541825208652762611246, −6.28396249907723201603327899088, −5.27461904287277902549120057686, −2.98992942803779082318067691672, −2.41218186200037421504477115783, −1.28991080725586359721882999750, −0.18475467836405768476338680412,
2.43072010846673202855284819410, 3.47725811711752918898264119339, 4.75152295609782881031287881609, 5.51719684595395905630295848273, 6.46913417080508007443984357826, 8.162413412389258628832969670734, 8.453174926825239788858410364217, 10.06799090164732486957934654880, 10.43600209282465574285677905962, 11.10012890585024235559798136618