| L(s) = 1 | + (8 + 13.8i)2-s + (38.4 − 66.5i)3-s + (−127. + 221. i)4-s + (546. + 945. i)5-s + 1.22e3·6-s + (6.11e3 + 1.73e3i)7-s − 4.09e3·8-s + (6.88e3 + 1.19e4i)9-s + (−8.73e3 + 1.51e4i)10-s + (−2.70e4 + 4.68e4i)11-s + (9.83e3 + 1.70e4i)12-s + 3.78e4·13-s + (2.48e4 + 9.85e4i)14-s + 8.39e4·15-s + (−3.27e4 − 5.67e4i)16-s + (1.76e5 − 3.06e5i)17-s + ⋯ |
| L(s) = 1 | + (0.353 + 0.612i)2-s + (0.273 − 0.474i)3-s + (−0.249 + 0.433i)4-s + (0.390 + 0.676i)5-s + 0.387·6-s + (0.962 + 0.272i)7-s − 0.353·8-s + (0.349 + 0.606i)9-s + (−0.276 + 0.478i)10-s + (−0.556 + 0.964i)11-s + (0.136 + 0.237i)12-s + 0.367·13-s + (0.173 + 0.685i)14-s + 0.428·15-s + (−0.125 − 0.216i)16-s + (0.513 − 0.888i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.332 - 0.942i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.332 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.89166 + 1.33810i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.89166 + 1.33810i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-8 - 13.8i)T \) |
| 7 | \( 1 + (-6.11e3 - 1.73e3i)T \) |
| good | 3 | \( 1 + (-38.4 + 66.5i)T + (-9.84e3 - 1.70e4i)T^{2} \) |
| 5 | \( 1 + (-546. - 945. i)T + (-9.76e5 + 1.69e6i)T^{2} \) |
| 11 | \( 1 + (2.70e4 - 4.68e4i)T + (-1.17e9 - 2.04e9i)T^{2} \) |
| 13 | \( 1 - 3.78e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + (-1.76e5 + 3.06e5i)T + (-5.92e10 - 1.02e11i)T^{2} \) |
| 19 | \( 1 + (2.51e4 + 4.34e4i)T + (-1.61e11 + 2.79e11i)T^{2} \) |
| 23 | \( 1 + (1.22e6 + 2.12e6i)T + (-9.00e11 + 1.55e12i)T^{2} \) |
| 29 | \( 1 - 1.40e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + (2.89e6 - 5.00e6i)T + (-1.32e13 - 2.28e13i)T^{2} \) |
| 37 | \( 1 + (2.72e6 + 4.72e6i)T + (-6.49e13 + 1.12e14i)T^{2} \) |
| 41 | \( 1 + 1.73e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.43e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + (1.52e7 + 2.63e7i)T + (-5.59e14 + 9.69e14i)T^{2} \) |
| 53 | \( 1 + (-2.15e6 + 3.73e6i)T + (-1.64e15 - 2.85e15i)T^{2} \) |
| 59 | \( 1 + (-3.30e7 + 5.72e7i)T + (-4.33e15 - 7.50e15i)T^{2} \) |
| 61 | \( 1 + (-1.00e8 - 1.74e8i)T + (-5.84e15 + 1.01e16i)T^{2} \) |
| 67 | \( 1 + (-1.36e8 + 2.36e8i)T + (-1.36e16 - 2.35e16i)T^{2} \) |
| 71 | \( 1 - 1.43e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + (-1.84e8 + 3.18e8i)T + (-2.94e16 - 5.09e16i)T^{2} \) |
| 79 | \( 1 + (2.31e8 + 4.01e8i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 + 7.47e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + (-5.63e8 - 9.76e8i)T + (-1.75e17 + 3.03e17i)T^{2} \) |
| 97 | \( 1 + 1.27e8T + 7.60e17T^{2} \) |
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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
−17.96800679080578823838986189328, −16.17672008548012322223464338900, −14.70791415680969632152579938207, −13.83468451938482129881210223362, −12.33627582265913440622297757873, −10.42009798902072105414000861766, −8.219204014963705242011530924024, −6.93702263886968952486089283688, −4.94805609011934341831533276933, −2.25701673191906726057769793808,
1.32387113186369133833605957055, 3.78386950716898048662296284090, 5.49882248627150216124032924102, 8.381416905889511643979105141567, 9.908908623928403674539422108107, 11.37462258974041157039367478786, 12.99183584292842227296956941455, 14.22190624394328208389026808002, 15.61225587394821155318010873558, 17.22831203791343874214841040543
