L(s) = 1 | + (0.707 − 0.707i)2-s − 1.00i·4-s + (−0.707 − 0.707i)7-s + (−0.707 − 0.707i)8-s + 3.41i·11-s + (−1.43 + 1.43i)13-s − 1.00·14-s − 1.00·16-s + (1.54 − 1.54i)17-s + 2.04i·19-s + (2.41 + 2.41i)22-s + (1.15 + 1.15i)23-s + 2.03i·26-s + (−0.707 + 0.707i)28-s − 8.04·29-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s − 0.500i·4-s + (−0.267 − 0.267i)7-s + (−0.250 − 0.250i)8-s + 1.02i·11-s + (−0.399 + 0.399i)13-s − 0.267·14-s − 0.250·16-s + (0.374 − 0.374i)17-s + 0.470i·19-s + (0.514 + 0.514i)22-s + (0.241 + 0.241i)23-s + 0.399i·26-s + (−0.133 + 0.133i)28-s − 1.49·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.161 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.161 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.101112827\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.101112827\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + (0.707 + 0.707i)T \) |
good | 11 | \( 1 - 3.41iT - 11T^{2} \) |
| 13 | \( 1 + (1.43 - 1.43i)T - 13iT^{2} \) |
| 17 | \( 1 + (-1.54 + 1.54i)T - 17iT^{2} \) |
| 19 | \( 1 - 2.04iT - 19T^{2} \) |
| 23 | \( 1 + (-1.15 - 1.15i)T + 23iT^{2} \) |
| 29 | \( 1 + 8.04T + 29T^{2} \) |
| 31 | \( 1 + 6.37T + 31T^{2} \) |
| 37 | \( 1 + (-0.0498 - 0.0498i)T + 37iT^{2} \) |
| 41 | \( 1 - 7.94iT - 41T^{2} \) |
| 43 | \( 1 + (4.65 - 4.65i)T - 43iT^{2} \) |
| 47 | \( 1 + (-1.39 + 1.39i)T - 47iT^{2} \) |
| 53 | \( 1 + (2.97 + 2.97i)T + 53iT^{2} \) |
| 59 | \( 1 - 4.25T + 59T^{2} \) |
| 61 | \( 1 + 6.88T + 61T^{2} \) |
| 67 | \( 1 + (-5.84 - 5.84i)T + 67iT^{2} \) |
| 71 | \( 1 + 2.92iT - 71T^{2} \) |
| 73 | \( 1 + (5.97 - 5.97i)T - 73iT^{2} \) |
| 79 | \( 1 - 11.6iT - 79T^{2} \) |
| 83 | \( 1 + (-1.52 - 1.52i)T + 83iT^{2} \) |
| 89 | \( 1 + 5.72T + 89T^{2} \) |
| 97 | \( 1 + (-9.64 - 9.64i)T + 97iT^{2} \) |
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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
−9.170196905868129002741076947217, −7.965106652824429205399763790257, −7.26861748765265049622417940446, −6.62590667925157960756763261603, −5.62391724583812188210496642185, −4.94161402063551015516496708261, −4.10237544390611469430803970383, −3.36210693942609757049395135595, −2.29638417366889197514547395971, −1.40767654736903383798807150691,
0.27258980048435326874109320390, 1.97656019902731778130060265939, 3.13947930292168024544584912797, 3.68469200602485757445996139347, 4.78861445182784874009299199681, 5.64560420751193844219928764295, 5.99876437068533637813739559891, 7.08853891962231079629593569027, 7.57289243665469380865546650504, 8.553130749956238900247871506683