L(s) = 1 | + 1.41i·3-s + 5-s + 7-s − 1.00·9-s − 11-s + 1.41i·15-s + 17-s + 1.41i·21-s − 1.41i·29-s − 1.41i·33-s + 35-s + 1.41i·37-s − 1.41i·41-s − 43-s − 1.00·45-s + ⋯ |
L(s) = 1 | + 1.41i·3-s + 5-s + 7-s − 1.00·9-s − 11-s + 1.41i·15-s + 17-s + 1.41i·21-s − 1.41i·29-s − 1.41i·33-s + 35-s + 1.41i·37-s − 1.41i·41-s − 43-s − 1.00·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.375162888\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.375162888\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
good | 3 | \( 1 - 1.41iT - T^{2} \) |
| 5 | \( 1 - T + T^{2} \) |
| 7 | \( 1 - T + T^{2} \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T + T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + 1.41iT - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - 1.41iT - T^{2} \) |
| 41 | \( 1 + 1.41iT - T^{2} \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( 1 + T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + 1.41iT - T^{2} \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - 1.41iT - T^{2} \) |
| 73 | \( 1 - T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.982688477019019676449701848876, −9.383922560218654592964385778835, −8.327991113120855842680468721897, −7.78092285616281715056948987331, −6.39945860036386387640003245147, −5.31016747309092775443413896561, −5.12110165769165580279851583405, −4.05794158105469510604608631770, −2.96572444043556335435036459743, −1.80070858114653563976834752223,
1.36050268332993543305465509124, 2.03785135530993626435906984304, 3.10830175164390682458008859562, 4.82467906905351532808468452381, 5.54853898334982717113260319787, 6.27654646058835998443996530083, 7.26487135219802101101900613453, 7.84055444481413429844057499486, 8.478877344675692620429186750915, 9.541781335649456715839484153441