L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.707 + 0.707i)3-s + (0.499 − 0.866i)4-s + (−0.258 − 0.965i)5-s + (−0.965 − 0.258i)6-s + (0.965 + 0.258i)7-s + 0.999i·8-s + 1.00i·9-s + (0.707 + 0.707i)10-s + (0.866 − 0.5i)11-s + (0.965 − 0.258i)12-s + (0.258 + 0.965i)13-s + (−0.965 + 0.258i)14-s + (0.500 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.707 − 0.707i)17-s + ⋯ |
L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.707 + 0.707i)3-s + (0.499 − 0.866i)4-s + (−0.258 − 0.965i)5-s + (−0.965 − 0.258i)6-s + (0.965 + 0.258i)7-s + 0.999i·8-s + 1.00i·9-s + (0.707 + 0.707i)10-s + (0.866 − 0.5i)11-s + (0.965 − 0.258i)12-s + (0.258 + 0.965i)13-s + (−0.965 + 0.258i)14-s + (0.500 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.707 − 0.707i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.784 - 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.784 - 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9962475565\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9962475565\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 3 | \( 1 + (-0.707 - 0.707i)T \) |
| 5 | \( 1 + (0.258 + 0.965i)T \) |
| 7 | \( 1 + (-0.965 - 0.258i)T \) |
good | 11 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \) |
| 17 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 19 | \( 1 + 1.41iT - T^{2} \) |
| 23 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \) |
| 47 | \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 + (-1 - i)T + iT^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 - iT - T^{2} \) |
| 73 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
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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
−9.456669799165742500660467674796, −8.922025287821713541004954565784, −8.761493022809697199942068848560, −7.77464486672265413957386389420, −6.96709684042520236834997327831, −5.74690404092718097128593583509, −4.76443947183303500353414700236, −4.25502980836418021619378812456, −2.55466819829605733558901816537, −1.38574784791730460683208968310,
1.43376453720233065914952020739, 2.28345370792543775389751610817, 3.47124582877086611054757854744, 4.10825238949672578719474216880, 6.05353084339144217988261542357, 6.82842568405509370094671586936, 7.67199109413722563118830637519, 8.068331177677659740650744540634, 8.848774016254985449256397775391, 9.841846124023789749029586742711