| L(s) = 1 | − 2-s + 4-s + (−2.23 + 1.41i)7-s − 8-s + 1.41i·11-s − 5.39·13-s + (2.23 − 1.41i)14-s + 16-s − 2.23i·17-s + 1.30i·19-s − 1.41i·22-s − 23-s + 5.39·26-s + (−2.23 + 1.41i)28-s + 9.24i·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + (−0.845 + 0.534i)7-s − 0.353·8-s + 0.426i·11-s − 1.49·13-s + (0.597 − 0.377i)14-s + 0.250·16-s − 0.542i·17-s + 0.300i·19-s − 0.301i·22-s − 0.208·23-s + 1.05·26-s + (−0.422 + 0.267i)28-s + 1.71i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.656 + 0.754i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.656 + 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6613537780\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6613537780\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.23 - 1.41i)T \) |
| good | 11 | \( 1 - 1.41iT - 11T^{2} \) |
| 13 | \( 1 + 5.39T + 13T^{2} \) |
| 17 | \( 1 + 2.23iT - 17T^{2} \) |
| 19 | \( 1 - 1.30iT - 19T^{2} \) |
| 23 | \( 1 + T + 23T^{2} \) |
| 29 | \( 1 - 9.24iT - 29T^{2} \) |
| 31 | \( 1 + 8.56iT - 31T^{2} \) |
| 37 | \( 1 + 2.82iT - 37T^{2} \) |
| 41 | \( 1 + 4.08T + 41T^{2} \) |
| 43 | \( 1 + 6.41iT - 43T^{2} \) |
| 47 | \( 1 - 7.63iT - 47T^{2} \) |
| 53 | \( 1 - 6.07T + 53T^{2} \) |
| 59 | \( 1 - 11.7T + 59T^{2} \) |
| 61 | \( 1 + 5.39iT - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 4.34iT - 71T^{2} \) |
| 73 | \( 1 + 5.01T + 73T^{2} \) |
| 79 | \( 1 + 1.07T + 79T^{2} \) |
| 83 | \( 1 + 8.01iT - 83T^{2} \) |
| 89 | \( 1 - 15.2T + 89T^{2} \) |
| 97 | \( 1 - 18.4T + 97T^{2} \) |
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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
−8.815442216878316027236874902709, −7.71378036664053121502587312307, −7.22801380789945834350855806569, −6.51044830494240808813427874865, −5.59859165012483129956968771254, −4.86576560254860728955438031821, −3.66819628203310362121817134687, −2.69294376489547866044025259639, −1.99097960309358159841943461110, −0.35889591257301730401713236085,
0.76125672152594412186332497541, 2.18018595585389559316627035009, 3.03925044760253783128288857005, 3.97494674627272059879656695965, 4.98587823893252845716648431606, 5.95227652763346574707640610176, 6.74198194859854957768089154789, 7.26524646419629485788594199377, 8.088349347316859775973973270391, 8.777498189208152192420604760553
