| L(s) = 1 | − i·2-s − i·3-s − 4-s + 4.05i·5-s − 6-s + i·8-s − 9-s + 4.05·10-s + (−1.17 − 3.09i)11-s + i·12-s − 3.91·13-s + 4.05·15-s + 16-s + 0.188·17-s + i·18-s + 4.37·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.577i·3-s − 0.5·4-s + 1.81i·5-s − 0.408·6-s + 0.353i·8-s − 0.333·9-s + 1.28·10-s + (−0.355 − 0.934i)11-s + 0.288i·12-s − 1.08·13-s + 1.04·15-s + 0.250·16-s + 0.0456·17-s + 0.235i·18-s + 1.00·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0579 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0579 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.225916829\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.225916829\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + iT \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (1.17 + 3.09i)T \) |
| good | 5 | \( 1 - 4.05iT - 5T^{2} \) |
| 13 | \( 1 + 3.91T + 13T^{2} \) |
| 17 | \( 1 - 0.188T + 17T^{2} \) |
| 19 | \( 1 - 4.37T + 19T^{2} \) |
| 23 | \( 1 + 3.58T + 23T^{2} \) |
| 29 | \( 1 + 5.75iT - 29T^{2} \) |
| 31 | \( 1 + 3.21iT - 31T^{2} \) |
| 37 | \( 1 - 5.09T + 37T^{2} \) |
| 41 | \( 1 - 8.14T + 41T^{2} \) |
| 43 | \( 1 + 3.21iT - 43T^{2} \) |
| 47 | \( 1 - 4.37iT - 47T^{2} \) |
| 53 | \( 1 - 2.83T + 53T^{2} \) |
| 59 | \( 1 - 4.99iT - 59T^{2} \) |
| 61 | \( 1 + 7.95T + 61T^{2} \) |
| 67 | \( 1 + 6.50T + 67T^{2} \) |
| 71 | \( 1 - 16.2T + 71T^{2} \) |
| 73 | \( 1 - 12.3T + 73T^{2} \) |
| 79 | \( 1 - 8.55iT - 79T^{2} \) |
| 83 | \( 1 - 12.5T + 83T^{2} \) |
| 89 | \( 1 + 14.2iT - 89T^{2} \) |
| 97 | \( 1 + 17.4iT - 97T^{2} \) |
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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
−8.268680846931926348104129320440, −7.63479047710109408998947228310, −7.15064251292771851878637027732, −6.09189663204241394507791151235, −5.70201500247307963280043747019, −4.35291161429178386032462958468, −3.34771390819962589750737395423, −2.72439797079766115259601419563, −2.11928167464865223947715088212, −0.47340017614315137986325491137,
0.906170200141169890920293789237, 2.21995690647134794299515675660, 3.66870339520612054520559304031, 4.57610199527521523265317344231, 5.03094889664482291740142574551, 5.46817442885612276342813868391, 6.56757941699685006235118526084, 7.75016041885200860785042392028, 7.87423756078794376450006524169, 8.990065296538130310576887269576
