L(s) = 1 | + (0.707 − 2.12i)5-s + (−2 + 2i)7-s − 2.82i·11-s + (−3 − 3i)13-s + (2.82 + 2.82i)17-s − 8i·19-s + (2.82 − 2.82i)23-s + (−3.99 − 3i)25-s − 1.41·29-s − 4·31-s + (2.82 + 5.65i)35-s + (−3 + 3i)37-s − 9.89i·41-s − i·49-s + (2.82 − 2.82i)53-s + ⋯ |
L(s) = 1 | + (0.316 − 0.948i)5-s + (−0.755 + 0.755i)7-s − 0.852i·11-s + (−0.832 − 0.832i)13-s + (0.685 + 0.685i)17-s − 1.83i·19-s + (0.589 − 0.589i)23-s + (−0.799 − 0.600i)25-s − 0.262·29-s − 0.718·31-s + (0.478 + 0.956i)35-s + (−0.493 + 0.493i)37-s − 1.54i·41-s − 0.142i·49-s + (0.388 − 0.388i)53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.256 + 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.256 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.679098 - 0.883229i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.679098 - 0.883229i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.707 + 2.12i)T \) |
good | 7 | \( 1 + (2 - 2i)T - 7iT^{2} \) |
| 11 | \( 1 + 2.82iT - 11T^{2} \) |
| 13 | \( 1 + (3 + 3i)T + 13iT^{2} \) |
| 17 | \( 1 + (-2.82 - 2.82i)T + 17iT^{2} \) |
| 19 | \( 1 + 8iT - 19T^{2} \) |
| 23 | \( 1 + (-2.82 + 2.82i)T - 23iT^{2} \) |
| 29 | \( 1 + 1.41T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + (3 - 3i)T - 37iT^{2} \) |
| 41 | \( 1 + 9.89iT - 41T^{2} \) |
| 43 | \( 1 + 43iT^{2} \) |
| 47 | \( 1 + 47iT^{2} \) |
| 53 | \( 1 + (-2.82 + 2.82i)T - 53iT^{2} \) |
| 59 | \( 1 + 2.82T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + (-4 + 4i)T - 67iT^{2} \) |
| 71 | \( 1 + 5.65iT - 71T^{2} \) |
| 73 | \( 1 + (-1 - i)T + 73iT^{2} \) |
| 79 | \( 1 - 12iT - 79T^{2} \) |
| 83 | \( 1 + (-8.48 + 8.48i)T - 83iT^{2} \) |
| 89 | \( 1 - 9.89T + 89T^{2} \) |
| 97 | \( 1 + (3 - 3i)T - 97iT^{2} \) |
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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
−10.06175428859333260412744522809, −9.104383264685273302016539805279, −8.737940774898491118809369137875, −7.64649409153445217082848329660, −6.48386041513352095327815771761, −5.55212560735151967607077628611, −4.94535050928679117811192896678, −3.44457019639751146849055941741, −2.38835102443905046876701905465, −0.56302754292156929325041158031,
1.81060055025570687130074595257, 3.13827739608353071299758350718, 4.03841521061702613344557650018, 5.34171549202662570852636868440, 6.42289560724105188641275513271, 7.21359377810603002845865786911, 7.68487209121867664774924762344, 9.332736932452204331552030681030, 9.875222527298685859087648160210, 10.40009009575296759388701286730