L(s) = 1 | + (−1 − i)5-s − i·13-s + i·25-s − 2·29-s + (1 − i)37-s + (−1 − i)41-s + i·49-s − 2·61-s + (−1 + i)65-s + (−1 + i)73-s + (1 − i)89-s + (−1 − i)97-s − 2i·101-s + (−1 − i)109-s − 2·113-s + ⋯ |
L(s) = 1 | + (−1 − i)5-s − i·13-s + i·25-s − 2·29-s + (1 − i)37-s + (−1 − i)41-s + i·49-s − 2·61-s + (−1 + i)65-s + (−1 + i)73-s + (1 − i)89-s + (−1 − i)97-s − 2i·101-s + (−1 − i)109-s − 2·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5643688942\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5643688942\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + iT \) |
good | 5 | \( 1 + (1 + i)T + iT^{2} \) |
| 7 | \( 1 - iT^{2} \) |
| 11 | \( 1 - iT^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + iT^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + 2T + T^{2} \) |
| 31 | \( 1 + iT^{2} \) |
| 37 | \( 1 + (-1 + i)T - iT^{2} \) |
| 41 | \( 1 + (1 + i)T + iT^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 - iT^{2} \) |
| 61 | \( 1 + 2T + T^{2} \) |
| 67 | \( 1 + iT^{2} \) |
| 71 | \( 1 + iT^{2} \) |
| 73 | \( 1 + (1 - i)T - iT^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + (-1 + i)T - iT^{2} \) |
| 97 | \( 1 + (1 + i)T + iT^{2} \) |
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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
−8.339742374672961279165232031769, −7.67500869256278450943271833205, −7.22151926179195700398412027215, −5.93036227048239425921511669264, −5.39546428376376608471313126149, −4.47937228504691894459332898697, −3.85431070512331280567216219998, −2.95321525711208065581558515371, −1.61508144026907618671371936935, −0.32072359653524332973986674631,
1.67573917283357882291012161368, 2.80431432439921721604670244259, 3.61202921888656863601696788705, 4.25487499357865804268424525686, 5.18708685007563075831132863235, 6.30573968906806554271417082287, 6.78302051541003112253828766859, 7.60081050113357637131959733652, 8.016814986215537215836916690020, 9.057932710982769147943482028470