L(s) = 1 | + (0.707 + 0.707i)3-s + (−2.49 − 0.884i)7-s + 1.00i·9-s + 3.27·11-s + (3.02 + 3.02i)13-s + (−1.41 + 1.41i)17-s + 2.15·19-s + (−1.13 − 2.38i)21-s + (0.296 − 0.296i)23-s + (−0.707 + 0.707i)27-s − 0.725i·29-s − 1.31i·31-s + (2.31 + 2.31i)33-s + (−1.56 − 1.56i)37-s + 4.27i·39-s + ⋯ |
L(s) = 1 | + (0.408 + 0.408i)3-s + (−0.942 − 0.334i)7-s + 0.333i·9-s + 0.987·11-s + (0.838 + 0.838i)13-s + (−0.342 + 0.342i)17-s + 0.493·19-s + (−0.248 − 0.521i)21-s + (0.0617 − 0.0617i)23-s + (−0.136 + 0.136i)27-s − 0.134i·29-s − 0.235i·31-s + (0.403 + 0.403i)33-s + (−0.256 − 0.256i)37-s + 0.684i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.454 - 0.890i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.454 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.885886795\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.885886795\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.707 - 0.707i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.49 + 0.884i)T \) |
good | 11 | \( 1 - 3.27T + 11T^{2} \) |
| 13 | \( 1 + (-3.02 - 3.02i)T + 13iT^{2} \) |
| 17 | \( 1 + (1.41 - 1.41i)T - 17iT^{2} \) |
| 19 | \( 1 - 2.15T + 19T^{2} \) |
| 23 | \( 1 + (-0.296 + 0.296i)T - 23iT^{2} \) |
| 29 | \( 1 + 0.725iT - 29T^{2} \) |
| 31 | \( 1 + 1.31iT - 31T^{2} \) |
| 37 | \( 1 + (1.56 + 1.56i)T + 37iT^{2} \) |
| 41 | \( 1 - 5.25iT - 41T^{2} \) |
| 43 | \( 1 + (0.632 - 0.632i)T - 43iT^{2} \) |
| 47 | \( 1 + (1.41 - 1.41i)T - 47iT^{2} \) |
| 53 | \( 1 + (3.71 - 3.71i)T - 53iT^{2} \) |
| 59 | \( 1 - 8.71T + 59T^{2} \) |
| 61 | \( 1 - 1.31iT - 61T^{2} \) |
| 67 | \( 1 + (-3.08 - 3.08i)T + 67iT^{2} \) |
| 71 | \( 1 - 9.27T + 71T^{2} \) |
| 73 | \( 1 + (-7.07 - 7.07i)T + 73iT^{2} \) |
| 79 | \( 1 - 11.8iT - 79T^{2} \) |
| 83 | \( 1 + (-11.7 - 11.7i)T + 83iT^{2} \) |
| 89 | \( 1 - 18.2T + 89T^{2} \) |
| 97 | \( 1 + (7.26 - 7.26i)T - 97iT^{2} \) |
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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
−9.335481101055020064060001039369, −8.642908297983102604461500568844, −7.75812204874912724907459229589, −6.65061618741108337989618046289, −6.39545192351816931689240648353, −5.18728355635436401637073230815, −3.96621082198457660141890326036, −3.74291923884824119664698642667, −2.51571092866260542478026562458, −1.19867316777442400001090560676,
0.72372538047771987225335674474, 2.03057293818117088606252195446, 3.24138385254813964759985973858, 3.67715866759217842491663340305, 5.03044193271499353933233141935, 6.01896173957169838701989284386, 6.59858989655540541312307355506, 7.34940212150290891624179748012, 8.315245900636020990351163492761, 8.963424397472675387568776838251