L(s) = 1 | − 3-s + (−2 + i)5-s + 4i·7-s + 9-s − 4·13-s + (2 − i)15-s − 8i·19-s − 4i·21-s + 4i·23-s + (3 − 4i)25-s − 27-s − 6i·29-s − 8·31-s + (−4 − 8i)35-s + 4·37-s + ⋯ |
L(s) = 1 | − 0.577·3-s + (−0.894 + 0.447i)5-s + 1.51i·7-s + 0.333·9-s − 1.10·13-s + (0.516 − 0.258i)15-s − 1.83i·19-s − 0.872i·21-s + 0.834i·23-s + (0.600 − 0.800i)25-s − 0.192·27-s − 1.11i·29-s − 1.43·31-s + (−0.676 − 1.35i)35-s + 0.657·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 + 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7008461913\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7008461913\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + (2 - i)T \) |
good | 7 | \( 1 - 4iT - 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 8iT - 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 + 12T + 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 - 6iT - 61T^{2} \) |
| 67 | \( 1 - 12T + 67T^{2} \) |
| 71 | \( 1 - 16T + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 8iT - 97T^{2} \) |
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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
−8.382882955059485389884885445931, −7.65179041888089091903267536187, −6.96864651613684846640506309669, −6.28682785761867600728419569781, −5.29928460055269320164416411745, −4.90736591779693756590007268187, −3.84259030884963158313517570022, −2.80359073554805024855614781734, −2.15302621110122197892159420102, −0.34893836721999059145476666389,
0.68374610464761706226317329659, 1.75018420922452246462785580086, 3.37859797357040281767691114960, 3.96741458280904981083644037553, 4.71352881870770466914269786172, 5.34601859187613630728697069488, 6.48411129134155836823899061613, 7.13165750885617325852098939174, 7.76021848206766050725454005828, 8.221805290733080167403690067255