L(s) = 1 | + (0.965 + 0.258i)3-s + (0.258 − 0.965i)5-s + (−0.5 − 0.866i)7-s + (0.866 + 0.499i)9-s + (−0.965 − 1.67i)11-s + (0.499 − 0.866i)15-s + (−0.258 − 0.965i)21-s + (−0.866 − 0.499i)25-s + (0.707 + 0.707i)27-s + 0.517i·29-s + (−0.866 + 0.5i)31-s + (−0.500 − 1.86i)33-s + (−0.965 + 0.258i)35-s + (0.707 − 0.707i)45-s + (−0.499 + 0.866i)49-s + ⋯ |
L(s) = 1 | + (0.965 + 0.258i)3-s + (0.258 − 0.965i)5-s + (−0.5 − 0.866i)7-s + (0.866 + 0.499i)9-s + (−0.965 − 1.67i)11-s + (0.499 − 0.866i)15-s + (−0.258 − 0.965i)21-s + (−0.866 − 0.499i)25-s + (0.707 + 0.707i)27-s + 0.517i·29-s + (−0.866 + 0.5i)31-s + (−0.500 − 1.86i)33-s + (−0.965 + 0.258i)35-s + (0.707 − 0.707i)45-s + (−0.499 + 0.866i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.104 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.104 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.573842322\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.573842322\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.965 - 0.258i)T \) |
| 5 | \( 1 + (-0.258 + 0.965i)T \) |
| 7 | \( 1 + (0.5 + 0.866i)T \) |
good | 11 | \( 1 + (0.965 + 1.67i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 29 | \( 1 - 0.517iT - T^{2} \) |
| 31 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 + (-0.448 + 1.67i)T + (-0.866 - 0.5i)T^{2} \) |
| 59 | \( 1 + (0.258 + 0.448i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \) |
| 79 | \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-1.22 + 1.22i)T - iT^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (0.366 + 0.366i)T + iT^{2} \) |
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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
−8.557518964029896158165185964706, −8.110508024907886908510552932611, −7.35431413033143446399165725255, −6.39992772450889270292002691745, −5.43247070486977033452753822639, −4.79434414415351258496190131896, −3.70163482937402343485150340660, −3.26349731453363348742509862867, −2.08464328978932149115721015576, −0.799610821372171449831650195660,
1.99577722945864124262363442249, 2.40294787868355227456252195843, 3.21427201598214020639872012401, 4.17581452072021852754084353528, 5.20781431080303619109979460282, 6.14191083255355308428321283532, 6.88515671903839224587956384903, 7.53336596593334667139826339013, 8.041858890164837681134151465606, 9.201738263700911130612151378210