| L(s) = 1 | + (0.694 + 3.93i)2-s + (−4.94 + 28.0i)3-s + (−15.0 + 5.47i)4-s + (9.88 + 8.29i)5-s − 113.·6-s + (−125. − 105. i)7-s + (−32 − 55.4i)8-s + (−534. − 194. i)9-s + (−25.7 + 44.6i)10-s + (343. + 595. i)11-s + (−79.1 − 448. i)12-s + (224. − 81.7i)13-s + (328. − 568. i)14-s + (−281. + 236. i)15-s + (196. − 164. i)16-s + (−144. − 52.4i)17-s + ⋯ |
| L(s) = 1 | + (0.122 + 0.696i)2-s + (−0.317 + 1.79i)3-s + (−0.469 + 0.171i)4-s + (0.176 + 0.148i)5-s − 1.29·6-s + (−0.969 − 0.813i)7-s + (−0.176 − 0.306i)8-s + (−2.19 − 0.800i)9-s + (−0.0815 + 0.141i)10-s + (0.856 + 1.48i)11-s + (−0.158 − 0.899i)12-s + (0.368 − 0.134i)13-s + (0.447 − 0.774i)14-s + (−0.323 + 0.271i)15-s + (0.191 − 0.160i)16-s + (−0.120 − 0.0440i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0689 + 0.997i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.0689 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.437612 - 0.468891i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.437612 - 0.468891i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.694 - 3.93i)T \) |
| 37 | \( 1 + (3.38e3 + 7.60e3i)T \) |
| good | 3 | \( 1 + (4.94 - 28.0i)T + (-228. - 83.1i)T^{2} \) |
| 5 | \( 1 + (-9.88 - 8.29i)T + (542. + 3.07e3i)T^{2} \) |
| 7 | \( 1 + (125. + 105. i)T + (2.91e3 + 1.65e4i)T^{2} \) |
| 11 | \( 1 + (-343. - 595. i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + (-224. + 81.7i)T + (2.84e5 - 2.38e5i)T^{2} \) |
| 17 | \( 1 + (144. + 52.4i)T + (1.08e6 + 9.12e5i)T^{2} \) |
| 19 | \( 1 + (-196. + 1.11e3i)T + (-2.32e6 - 8.46e5i)T^{2} \) |
| 23 | \( 1 + (2.39e3 - 4.14e3i)T + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + (-1.13e3 - 1.96e3i)T + (-1.02e7 + 1.77e7i)T^{2} \) |
| 31 | \( 1 + 5.95e3T + 2.86e7T^{2} \) |
| 41 | \( 1 + (6.90e3 - 2.51e3i)T + (8.87e7 - 7.44e7i)T^{2} \) |
| 43 | \( 1 - 1.25e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-6.03e3 + 1.04e4i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + (2.44e4 - 2.04e4i)T + (7.26e7 - 4.11e8i)T^{2} \) |
| 59 | \( 1 + (3.07e4 - 2.58e4i)T + (1.24e8 - 7.04e8i)T^{2} \) |
| 61 | \( 1 + (1.42e4 - 5.19e3i)T + (6.46e8 - 5.42e8i)T^{2} \) |
| 67 | \( 1 + (-3.13e3 - 2.62e3i)T + (2.34e8 + 1.32e9i)T^{2} \) |
| 71 | \( 1 + (9.07e3 - 5.14e4i)T + (-1.69e9 - 6.17e8i)T^{2} \) |
| 73 | \( 1 + 3.33e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + (2.07e4 + 1.73e4i)T + (5.34e8 + 3.03e9i)T^{2} \) |
| 83 | \( 1 + (-8.08e4 - 2.94e4i)T + (3.01e9 + 2.53e9i)T^{2} \) |
| 89 | \( 1 + (-1.88e4 + 1.57e4i)T + (9.69e8 - 5.49e9i)T^{2} \) |
| 97 | \( 1 + (2.67e4 - 4.62e4i)T + (-4.29e9 - 7.43e9i)T^{2} \) |
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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
−14.67598934404775575316274696919, −13.69003576946908832640496077565, −12.14881889899967373987424967626, −10.69663732710246378073284477446, −9.755050956826009041671647401414, −9.178036689581210120605750084517, −7.16843580020007060506416257073, −5.87943451240273980772372004207, −4.45931824493571797116732724510, −3.61027657237724593171744214154,
0.27824021455065587873155687168, 1.70712803502281510959752094659, 3.18526410141688258478928668116, 5.87193327992905447276316127775, 6.40431573880212025529826507987, 8.206006096652599569391359653343, 9.158553391253143856024565085744, 10.98732833062446221811250956267, 12.00877315943106407245962455794, 12.61284846196343549729867174370
