L(s) = 1 | + (1 − i)2-s + (1.86 + 0.5i)3-s − 2i·4-s + (−0.133 + 2.23i)5-s + (2.36 − 1.36i)6-s + (0.866 − 2.5i)7-s + (−2 − 2i)8-s + (0.633 + 0.366i)9-s + (2.09 + 2.36i)10-s + (0.633 − 0.366i)11-s + (1 − 3.73i)12-s + (−3 + 3i)13-s + (−1.63 − 3.36i)14-s + (−1.36 + 4.09i)15-s − 4·16-s + (4.09 + 1.09i)17-s + ⋯ |
L(s) = 1 | + (0.707 − 0.707i)2-s + (1.07 + 0.288i)3-s − i·4-s + (−0.0599 + 0.998i)5-s + (0.965 − 0.557i)6-s + (0.327 − 0.944i)7-s + (−0.707 − 0.707i)8-s + (0.211 + 0.122i)9-s + (0.663 + 0.748i)10-s + (0.191 − 0.110i)11-s + (0.288 − 1.07i)12-s + (−0.832 + 0.832i)13-s + (−0.436 − 0.899i)14-s + (−0.352 + 1.05i)15-s − 16-s + (0.993 + 0.266i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.730 + 0.683i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.730 + 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.20886 - 0.872017i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.20886 - 0.872017i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1 + i)T \) |
| 5 | \( 1 + (0.133 - 2.23i)T \) |
| 7 | \( 1 + (-0.866 + 2.5i)T \) |
good | 3 | \( 1 + (-1.86 - 0.5i)T + (2.59 + 1.5i)T^{2} \) |
| 11 | \( 1 + (-0.633 + 0.366i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (3 - 3i)T - 13iT^{2} \) |
| 17 | \( 1 + (-4.09 - 1.09i)T + (14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (-3 - 1.73i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.401 - 1.5i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + 4.46T + 29T^{2} \) |
| 31 | \( 1 + (8.83 - 5.09i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.73 + 6.46i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + 0.464iT - 41T^{2} \) |
| 43 | \( 1 + (-2.36 - 2.36i)T + 43iT^{2} \) |
| 47 | \( 1 + (-0.169 - 0.633i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (0.803 - 3i)T + (-45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (-7.73 + 4.46i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.96 + 8.59i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (3.86 + 1.03i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 - 15.4T + 71T^{2} \) |
| 73 | \( 1 + (2.16 - 8.09i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-11.1 - 6.46i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (4.56 - 4.56i)T - 83iT^{2} \) |
| 89 | \( 1 + (-8.59 + 14.8i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (6.26 - 6.26i)T - 97iT^{2} \) |
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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
−11.62988828876085233461159122883, −10.87186275066179878525744190781, −9.913760007070162298129485408246, −9.291416231105555313481487378202, −7.74755342338894154378820966290, −6.91478729484676280178054894713, −5.44865415767917462709124295649, −3.90394222983684546146223926867, −3.38019176946268871953115662208, −1.98976639421233728469804954924,
2.30885854148395111787779706013, 3.51373938271878839443148989183, 5.06651400597055332186517501619, 5.63524496723586120461980288714, 7.41270832456188885269580035708, 7.985806684140267551315960356137, 8.856150741884632480976051535263, 9.547962308936544748566146008221, 11.52553834261788530549363034143, 12.31252538635482516093107056128