| L(s) = 1 | + i·3-s + i·4-s − i·7-s − 9-s − 12-s + i·13-s − 16-s + (1 − i)19-s + 21-s − i·25-s − i·27-s + 28-s + (1 − i)31-s − i·36-s + (−1 + i)37-s + ⋯ |
| L(s) = 1 | + i·3-s + i·4-s − i·7-s − 9-s − 12-s + i·13-s − 16-s + (1 − i)19-s + 21-s − i·25-s − i·27-s + 28-s + (1 − i)31-s − i·36-s + (−1 + i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.289 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.289 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7387973879\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7387973879\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - iT \) |
| 7 | \( 1 + iT \) |
| 13 | \( 1 - iT \) |
| good | 2 | \( 1 - iT^{2} \) |
| 5 | \( 1 + iT^{2} \) |
| 11 | \( 1 + iT^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + (-1 + i)T - iT^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (-1 + i)T - iT^{2} \) |
| 37 | \( 1 + (1 - i)T - iT^{2} \) |
| 41 | \( 1 + iT^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - iT^{2} \) |
| 61 | \( 1 - 2iT - T^{2} \) |
| 67 | \( 1 + (1 + i)T + iT^{2} \) |
| 71 | \( 1 - iT^{2} \) |
| 73 | \( 1 + (1 + i)T + iT^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 - iT^{2} \) |
| 97 | \( 1 + (1 - i)T - iT^{2} \) |
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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
−12.00826836791814404038914493387, −11.45923478981213974867197829413, −10.42250754467157448455559719041, −9.482285475438337191842642683814, −8.585749389926845033220554350605, −7.53437737870792191497581785993, −6.50169676484319392567364220558, −4.76295225401934554504501923838, −4.05396996410010264502837718157, −2.88086612501724707981685940959,
1.56447021090035348653554707857, 3.01778581380916191492865196462, 5.30788341960039774781022168600, 5.77691315884255959793611820593, 6.92425560930610313759710909888, 8.067561367206620885196632300357, 9.021923136603591052231311787150, 10.05518373812994704720834907736, 11.11952540500382454648203509087, 12.02755067078045430069713503733
