| L(s) = 1 | + (0.284 − 1.97i)2-s + (−3.91 − 8.57i)3-s + (−3.83 − 1.12i)4-s + (−3.27 − 3.77i)5-s + (−18.0 + 5.31i)6-s + (13.7 + 8.83i)7-s + (−3.32 + 7.27i)8-s + (−40.5 + 46.7i)9-s + (−8.41 + 5.40i)10-s + (−1.47 − 10.2i)11-s + (5.36 + 37.3i)12-s + (−48.3 + 31.0i)13-s + (21.3 − 24.6i)14-s + (−19.5 + 42.8i)15-s + (13.4 + 8.65i)16-s + (−52.8 + 15.5i)17-s + ⋯ |
| L(s) = 1 | + (0.100 − 0.699i)2-s + (−0.753 − 1.65i)3-s + (−0.479 − 0.140i)4-s + (−0.292 − 0.337i)5-s + (−1.23 + 0.361i)6-s + (0.742 + 0.476i)7-s + (−0.146 + 0.321i)8-s + (−1.50 + 1.73i)9-s + (−0.266 + 0.170i)10-s + (−0.0404 − 0.281i)11-s + (0.129 + 0.897i)12-s + (−1.03 + 0.663i)13-s + (0.408 − 0.471i)14-s + (−0.337 + 0.738i)15-s + (0.210 + 0.135i)16-s + (−0.753 + 0.221i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.782 - 0.622i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.782 - 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0932418 + 0.0325503i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0932418 + 0.0325503i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.284 + 1.97i)T \) |
| 5 | \( 1 + (3.27 + 3.77i)T \) |
| 23 | \( 1 + (110. - 7.02i)T \) |
| good | 3 | \( 1 + (3.91 + 8.57i)T + (-17.6 + 20.4i)T^{2} \) |
| 7 | \( 1 + (-13.7 - 8.83i)T + (142. + 312. i)T^{2} \) |
| 11 | \( 1 + (1.47 + 10.2i)T + (-1.27e3 + 374. i)T^{2} \) |
| 13 | \( 1 + (48.3 - 31.0i)T + (912. - 1.99e3i)T^{2} \) |
| 17 | \( 1 + (52.8 - 15.5i)T + (4.13e3 - 2.65e3i)T^{2} \) |
| 19 | \( 1 + (-59.8 - 17.5i)T + (5.77e3 + 3.70e3i)T^{2} \) |
| 29 | \( 1 + (-238. + 70.0i)T + (2.05e4 - 1.31e4i)T^{2} \) |
| 31 | \( 1 + (-65.6 + 143. i)T + (-1.95e4 - 2.25e4i)T^{2} \) |
| 37 | \( 1 + (233. - 270. i)T + (-7.20e3 - 5.01e4i)T^{2} \) |
| 41 | \( 1 + (320. + 369. i)T + (-9.80e3 + 6.82e4i)T^{2} \) |
| 43 | \( 1 + (-53.6 - 117. i)T + (-5.20e4 + 6.00e4i)T^{2} \) |
| 47 | \( 1 + 126.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-153. - 98.5i)T + (6.18e4 + 1.35e5i)T^{2} \) |
| 59 | \( 1 + (548. - 352. i)T + (8.53e4 - 1.86e5i)T^{2} \) |
| 61 | \( 1 + (273. - 597. i)T + (-1.48e5 - 1.71e5i)T^{2} \) |
| 67 | \( 1 + (-0.389 + 2.70i)T + (-2.88e5 - 8.47e4i)T^{2} \) |
| 71 | \( 1 + (60.3 - 419. i)T + (-3.43e5 - 1.00e5i)T^{2} \) |
| 73 | \( 1 + (-49.4 - 14.5i)T + (3.27e5 + 2.10e5i)T^{2} \) |
| 79 | \( 1 + (349. - 224. i)T + (2.04e5 - 4.48e5i)T^{2} \) |
| 83 | \( 1 + (783. - 904. i)T + (-8.13e4 - 5.65e5i)T^{2} \) |
| 89 | \( 1 + (-281. - 616. i)T + (-4.61e5 + 5.32e5i)T^{2} \) |
| 97 | \( 1 + (-304. - 351. i)T + (-1.29e5 + 9.03e5i)T^{2} \) |
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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
−11.88202482410673540613107976571, −11.51365971850051389131604991181, −10.20790326518227113567980442083, −8.669000998207949926598224746844, −7.915626771655509693619802247860, −6.80461581095294229054951053189, −5.63872368570500878420190489237, −4.62057540309138417423121937870, −2.44834193798086579998594037271, −1.42385657924938227525295845370,
0.04439352935958085637945925798, 3.31416173557728654598514412663, 4.67047073447195819710942894555, 4.99314883833741079355519822633, 6.39524334656775538533662015472, 7.60113173728739860933875762004, 8.776732014466439897982159259442, 9.953821778527818948090358808804, 10.48413236185601926832822705102, 11.49947130112603195359582778786
