L(s) = 1 | + (0.707 − 0.707i)2-s − 1.00i·4-s + (0.366 + 0.366i)7-s + (−0.707 − 0.707i)8-s + 4.76i·11-s + (−3.46 + 3.46i)13-s + 0.517·14-s − 1.00·16-s + (−1.03 + 1.03i)17-s + 7.46i·19-s + (3.36 + 3.36i)22-s + (−1.41 − 1.41i)23-s + 4.89i·26-s + (0.366 − 0.366i)28-s − 6.31·29-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s − 0.500i·4-s + (0.138 + 0.138i)7-s + (−0.250 − 0.250i)8-s + 1.43i·11-s + (−0.960 + 0.960i)13-s + 0.138·14-s − 0.250·16-s + (−0.251 + 0.251i)17-s + 1.71i·19-s + (0.717 + 0.717i)22-s + (−0.294 − 0.294i)23-s + 0.960i·26-s + (0.0691 − 0.0691i)28-s − 1.17·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.587046278\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.587046278\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-0.366 - 0.366i)T + 7iT^{2} \) |
| 11 | \( 1 - 4.76iT - 11T^{2} \) |
| 13 | \( 1 + (3.46 - 3.46i)T - 13iT^{2} \) |
| 17 | \( 1 + (1.03 - 1.03i)T - 17iT^{2} \) |
| 19 | \( 1 - 7.46iT - 19T^{2} \) |
| 23 | \( 1 + (1.41 + 1.41i)T + 23iT^{2} \) |
| 29 | \( 1 + 6.31T + 29T^{2} \) |
| 31 | \( 1 - 6.66T + 31T^{2} \) |
| 37 | \( 1 + (-2.19 - 2.19i)T + 37iT^{2} \) |
| 41 | \( 1 + 4.62iT - 41T^{2} \) |
| 43 | \( 1 + (-1.26 + 1.26i)T - 43iT^{2} \) |
| 47 | \( 1 + (4.24 - 4.24i)T - 47iT^{2} \) |
| 53 | \( 1 + (-5.08 - 5.08i)T + 53iT^{2} \) |
| 59 | \( 1 + 4.52T + 59T^{2} \) |
| 61 | \( 1 - 9.46T + 61T^{2} \) |
| 67 | \( 1 + (-6.19 - 6.19i)T + 67iT^{2} \) |
| 71 | \( 1 + 15.4iT - 71T^{2} \) |
| 73 | \( 1 + (9.36 - 9.36i)T - 73iT^{2} \) |
| 79 | \( 1 - 3.46iT - 79T^{2} \) |
| 83 | \( 1 + (-1.46 - 1.46i)T + 83iT^{2} \) |
| 89 | \( 1 + 5.93T + 89T^{2} \) |
| 97 | \( 1 + (-13.5 - 13.5i)T + 97iT^{2} \) |
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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
−9.909497203266547165135419229266, −9.188100111294749919393969683523, −8.066972305737797121484982838998, −7.22401906554408921200183250932, −6.39917616749354757458899226766, −5.38750303816109615467094061474, −4.49570910410841189890190032571, −3.86526893820192050926517767123, −2.39859121767611385391105765859, −1.70440623496560762538131849006,
0.52989045453619554196041620153, 2.55321983511391091952032180108, 3.33327990843366005362523510335, 4.53173643196275149924772110189, 5.30426531764726958799187353079, 6.08218228645337886172655567981, 7.01661761284084757514628213386, 7.77422080409304042857905085265, 8.522194393168299046623009101555, 9.340885126872597045541195133711