L(s) = 1 | + (1.5 − 2.59i)2-s + (−1.5 − 2.59i)3-s + (−0.5 − 0.866i)4-s + (1.5 − 2.59i)5-s − 9·6-s + (−3.5 + 18.1i)7-s + 21·8-s + (−4.5 + 7.79i)9-s + (−4.5 − 7.79i)10-s + (7.5 + 12.9i)11-s + (−1.50 + 2.59i)12-s − 64·13-s + (42 + 36.3i)14-s − 9·15-s + (35.5 − 61.4i)16-s + (−42 − 72.7i)17-s + ⋯ |
L(s) = 1 | + (0.530 − 0.918i)2-s + (−0.288 − 0.499i)3-s + (−0.0625 − 0.108i)4-s + (0.134 − 0.232i)5-s − 0.612·6-s + (−0.188 + 0.981i)7-s + 0.928·8-s + (−0.166 + 0.288i)9-s + (−0.142 − 0.246i)10-s + (0.205 + 0.356i)11-s + (−0.0360 + 0.0625i)12-s − 1.36·13-s + (0.801 + 0.694i)14-s − 0.154·15-s + (0.554 − 0.960i)16-s + (−0.599 − 1.03i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.10777 - 0.736870i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.10777 - 0.736870i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.5 + 2.59i)T \) |
| 7 | \( 1 + (3.5 - 18.1i)T \) |
good | 2 | \( 1 + (-1.5 + 2.59i)T + (-4 - 6.92i)T^{2} \) |
| 5 | \( 1 + (-1.5 + 2.59i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-7.5 - 12.9i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + 64T + 2.19e3T^{2} \) |
| 17 | \( 1 + (42 + 72.7i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-8 + 13.8i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-42 + 72.7i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + 297T + 2.43e4T^{2} \) |
| 31 | \( 1 + (-126.5 - 219. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-158 + 273. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 - 360T + 6.89e4T^{2} \) |
| 43 | \( 1 - 26T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-15 + 25.9i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (181.5 + 314. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-7.5 - 12.9i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-59 + 102. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-185 - 320. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 342T + 3.57e5T^{2} \) |
| 73 | \( 1 + (181 + 313. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (233.5 - 404. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 - 477T + 5.71e5T^{2} \) |
| 89 | \( 1 + (453 - 784. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 - 503T + 9.12e5T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−17.58386982955827734240686480395, −16.30416758363011446028043546281, −14.60107513759248829817160017074, −13.00932163019732403024923293547, −12.28229638658006112017284905068, −11.20719166378544860614006310948, −9.361730012047625925671515567963, −7.24478076920553501392747320947, −5.03511410905422182392571092911, −2.45162046213851339139222966109,
4.37886213680069290113882929938, 6.10912565339455725894963758727, 7.51666512192279009581180184974, 9.873668919387526237337095414518, 11.07506946322079037619428719899, 13.11785989273716789651633747066, 14.40025286648667157123636101921, 15.27106905125728587197844724311, 16.67477650348304783612437507985, 17.23074733756588066166355793207