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
−9.541781335649456715839484153441, −8.478877344675692620429186750915, −7.84055444481413429844057499486, −7.26487135219802101101900613453, −6.27654646058835998443996530083, −5.54853898334982717113260319787, −4.82467906905351532808468452381, −3.10830175164390682458008859562, −2.03785135530993626435906984304, −1.36050268332993543305465509124,
1.80070858114653563976834752223, 2.96572444043556335435036459743, 4.05794158105469510604608631770, 5.12110165769165580279851583405, 5.31016747309092775443413896561, 6.39945860036386387640003245147, 7.78092285616281715056948987331, 8.327991113120855842680468721897, 9.383922560218654592964385778835, 9.982688477019019676449701848876