L(s) = 1 | − i·2-s − 4-s − 4.14i·5-s + (−0.717 + 2.54i)7-s + i·8-s − 4.14·10-s + (0.170 + 3.31i)11-s − 3.09·13-s + (2.54 + 0.717i)14-s + 16-s + 3.57·17-s − 3.91·19-s + 4.14i·20-s + (3.31 − 0.170i)22-s − 7.71·23-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.5·4-s − 1.85i·5-s + (−0.271 + 0.962i)7-s + 0.353i·8-s − 1.30·10-s + (0.0514 + 0.998i)11-s − 0.857·13-s + (0.680 + 0.191i)14-s + 0.250·16-s + 0.867·17-s − 0.898·19-s + 0.926i·20-s + (0.706 − 0.0363i)22-s − 1.60·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.221 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.221 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3046488913\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3046488913\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.717 - 2.54i)T \) |
| 11 | \( 1 + (-0.170 - 3.31i)T \) |
good | 5 | \( 1 + 4.14iT - 5T^{2} \) |
| 13 | \( 1 + 3.09T + 13T^{2} \) |
| 17 | \( 1 - 3.57T + 17T^{2} \) |
| 19 | \( 1 + 3.91T + 19T^{2} \) |
| 23 | \( 1 + 7.71T + 23T^{2} \) |
| 29 | \( 1 + 1.65iT - 29T^{2} \) |
| 31 | \( 1 - 9.23iT - 31T^{2} \) |
| 37 | \( 1 + 0.869T + 37T^{2} \) |
| 41 | \( 1 + 5.91T + 41T^{2} \) |
| 43 | \( 1 - 10.2iT - 43T^{2} \) |
| 47 | \( 1 + 6.76iT - 47T^{2} \) |
| 53 | \( 1 - 1.88T + 53T^{2} \) |
| 59 | \( 1 - 10.1iT - 59T^{2} \) |
| 61 | \( 1 - 3.37T + 61T^{2} \) |
| 67 | \( 1 - 9.84T + 67T^{2} \) |
| 71 | \( 1 + 1.33T + 71T^{2} \) |
| 73 | \( 1 - 5.57T + 73T^{2} \) |
| 79 | \( 1 + 10.8iT - 79T^{2} \) |
| 83 | \( 1 + 5.67T + 83T^{2} \) |
| 89 | \( 1 + 10.6iT - 89T^{2} \) |
| 97 | \( 1 - 4.52iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.899103567199641978545397905513, −8.943349895245749805895983799702, −8.458726477213084940720366882365, −7.63399282834979523983188375730, −6.22455923466731157675041519456, −5.22587957101964403606250921106, −4.77861083659562010508986314754, −3.81157972715587790423317427623, −2.34556015366105432655426939148, −1.50848150276732293359246757540,
0.11992921891845607502612358950, 2.32603081439317535632393651689, 3.48883630153611333815811814123, 4.03129632889021614792487598035, 5.54794347130440828644777457983, 6.33378450288950095537179788204, 6.88829888501366710812123948827, 7.67604181938723026516674295666, 8.207659402371730407523538079120, 9.685513115454782182988578393566