L(s) = 1 | + (0.382 − 0.923i)3-s + (0.541 − 0.541i)7-s + (−0.707 − 0.707i)9-s + (1.70 + 0.707i)13-s + (−1.30 − 0.541i)19-s + (−0.292 − 0.707i)21-s + (0.707 − 0.707i)25-s + (−0.923 + 0.382i)27-s + 1.84·31-s + (−0.707 + 0.292i)37-s + (1.30 − 1.30i)39-s + (−0.541 − 1.30i)43-s + 0.414i·49-s + (−1 + 0.999i)57-s + (−0.707 + 1.70i)61-s + ⋯ |
L(s) = 1 | + (0.382 − 0.923i)3-s + (0.541 − 0.541i)7-s + (−0.707 − 0.707i)9-s + (1.70 + 0.707i)13-s + (−1.30 − 0.541i)19-s + (−0.292 − 0.707i)21-s + (0.707 − 0.707i)25-s + (−0.923 + 0.382i)27-s + 1.84·31-s + (−0.707 + 0.292i)37-s + (1.30 − 1.30i)39-s + (−0.541 − 1.30i)43-s + 0.414i·49-s + (−1 + 0.999i)57-s + (−0.707 + 1.70i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.195 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.195 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.546384809\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.546384809\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.382 + 0.923i)T \) |
good | 5 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 7 | \( 1 + (-0.541 + 0.541i)T - iT^{2} \) |
| 11 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 13 | \( 1 + (-1.70 - 0.707i)T + (0.707 + 0.707i)T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + (1.30 + 0.541i)T + (0.707 + 0.707i)T^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 31 | \( 1 - 1.84T + T^{2} \) |
| 37 | \( 1 + (0.707 - 0.292i)T + (0.707 - 0.707i)T^{2} \) |
| 41 | \( 1 - iT^{2} \) |
| 43 | \( 1 + (0.541 + 1.30i)T + (-0.707 + 0.707i)T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 59 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 61 | \( 1 + (0.707 - 1.70i)T + (-0.707 - 0.707i)T^{2} \) |
| 67 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 71 | \( 1 + iT^{2} \) |
| 73 | \( 1 + (1 + i)T + iT^{2} \) |
| 79 | \( 1 + 1.84iT - T^{2} \) |
| 83 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 89 | \( 1 + iT^{2} \) |
| 97 | \( 1 - 1.41T + T^{2} \) |
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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
−8.635254787922849039787972318485, −8.109539461054794789270015396620, −7.17388694497541660879402764629, −6.49218553981150172575729028438, −6.04711232044242467508324236782, −4.70868297867080282857610122283, −4.00590364479828792775518949498, −2.98714584168916666603824810032, −1.92764796764071932154030066015, −1.02538880827801870979427363888,
1.52827055133606254980378963983, 2.72379420524008302338984415808, 3.52984534232785803063335439413, 4.34745715312598608404957164562, 5.15837647806033425390362341002, 5.91327057774597595842023209633, 6.62721595047507918918665091178, 8.053724219209599640982038412125, 8.331145441721883964650958857843, 8.877577586562685156155730356060