| 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
−12.02755067078045430069713503733, −11.11952540500382454648203509087, −10.05518373812994704720834907736, −9.021923136603591052231311787150, −8.067561367206620885196632300357, −6.92425560930610313759710909888, −5.77691315884255959793611820593, −5.30788341960039774781022168600, −3.01778581380916191492865196462, −1.56447021090035348653554707857,
2.88086612501724707981685940959, 4.05396996410010264502837718157, 4.76295225401934554504501923838, 6.50169676484319392567364220558, 7.53437737870792191497581785993, 8.585749389926845033220554350605, 9.482285475438337191842642683814, 10.42250754467157448455559719041, 11.45923478981213974867197829413, 12.00826836791814404038914493387
