| L(s) = 1 | + (1 − i)3-s + (−1 − 2i)5-s + (1 − i)7-s + i·9-s + 4·11-s + (−3 − 3i)13-s + (−3 − i)15-s + (−3 − 3i)17-s − 6i·19-s − 2i·21-s + (3 + 3i)23-s + (−3 + 4i)25-s + (4 + 4i)27-s + 2·29-s + 6i·31-s + ⋯ |
| L(s) = 1 | + (0.577 − 0.577i)3-s + (−0.447 − 0.894i)5-s + (0.377 − 0.377i)7-s + 0.333i·9-s + 1.20·11-s + (−0.832 − 0.832i)13-s + (−0.774 − 0.258i)15-s + (−0.727 − 0.727i)17-s − 1.37i·19-s − 0.436i·21-s + (0.625 + 0.625i)23-s + (−0.600 + 0.800i)25-s + (0.769 + 0.769i)27-s + 0.371·29-s + 1.07i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.17676 - 0.931316i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.17676 - 0.931316i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (1 + 2i)T \) |
| good | 3 | \( 1 + (-1 + i)T - 3iT^{2} \) |
| 7 | \( 1 + (-1 + i)T - 7iT^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + (3 + 3i)T + 13iT^{2} \) |
| 17 | \( 1 + (3 + 3i)T + 17iT^{2} \) |
| 19 | \( 1 + 6iT - 19T^{2} \) |
| 23 | \( 1 + (-3 - 3i)T + 23iT^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 6iT - 31T^{2} \) |
| 37 | \( 1 + (3 - 3i)T - 37iT^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + (3 - 3i)T - 43iT^{2} \) |
| 47 | \( 1 + (-9 + 9i)T - 47iT^{2} \) |
| 53 | \( 1 + (-5 - 5i)T + 53iT^{2} \) |
| 59 | \( 1 - 10iT - 59T^{2} \) |
| 61 | \( 1 - 12iT - 61T^{2} \) |
| 67 | \( 1 + (9 + 9i)T + 67iT^{2} \) |
| 71 | \( 1 - 6iT - 71T^{2} \) |
| 73 | \( 1 + (-5 + 5i)T - 73iT^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + (3 - 3i)T - 83iT^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + (7 + 7i)T + 97iT^{2} \) |
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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
−11.61430812631216105871510866690, −10.59988447635737109397568659638, −9.173714116762044901093328545790, −8.699215331916632323856246555226, −7.52852954767173224025226745630, −7.00410605261467062306638326401, −5.21359479119464999923726927145, −4.38548362701850026288655848623, −2.78602271721816750954710911455, −1.14701040917596716806492373676,
2.23191198852958141835641901402, 3.67802525611135460159399783198, 4.36018674895597463202904052663, 6.14282944022681839691336982259, 6.96678563256221610647661711931, 8.194058849569125180362096735563, 9.080884202459432893602604602317, 9.872085441494781428469158973204, 10.90084021176686885646320573268, 11.81113829924476911159363322084
