| L(s) = 1 | + (−1.41 − i)3-s + 1.41i·5-s + i·7-s + (1.00 + 2.82i)9-s − 1.41·11-s + 2·13-s + (1.41 − 2.00i)15-s + 7.07i·17-s + 4i·19-s + (1 − 1.41i)21-s + 7.07·23-s + 2.99·25-s + (1.41 − 5.00i)27-s − 2.82i·29-s + 6i·31-s + ⋯ |
| L(s) = 1 | + (−0.816 − 0.577i)3-s + 0.632i·5-s + 0.377i·7-s + (0.333 + 0.942i)9-s − 0.426·11-s + 0.554·13-s + (0.365 − 0.516i)15-s + 1.71i·17-s + 0.917i·19-s + (0.218 − 0.308i)21-s + 1.47·23-s + 0.599·25-s + (0.272 − 0.962i)27-s − 0.525i·29-s + 1.07i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.828172 + 0.428693i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.828172 + 0.428693i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.41 + i)T \) |
| 7 | \( 1 - iT \) |
| good | 5 | \( 1 - 1.41iT - 5T^{2} \) |
| 11 | \( 1 + 1.41T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 7.07iT - 17T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 - 7.07T + 23T^{2} \) |
| 29 | \( 1 + 2.82iT - 29T^{2} \) |
| 31 | \( 1 - 6iT - 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 + 1.41iT - 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 8.48T + 47T^{2} \) |
| 53 | \( 1 + 11.3iT - 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 - 7.07T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 + 12iT - 79T^{2} \) |
| 83 | \( 1 - 14.1T + 83T^{2} \) |
| 89 | \( 1 + 9.89iT - 89T^{2} \) |
| 97 | \( 1 - 14T + 97T^{2} \) |
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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
−11.63661032007667244157382455287, −10.75175412609563244681658219561, −10.28156442275347272255289199655, −8.714055131385387501005280130203, −7.82412276457851540367002629661, −6.69206176507461734165701215079, −6.01712197959749380778088466709, −4.92161903450473878356750756419, −3.30645263191911743306282309204, −1.67333750734329715104186146833,
0.78006210107705623576331751851, 3.16320364538281076917752848877, 4.69279535475897470802071833809, 5.15465076509097058619819270697, 6.52054732817154117551500702269, 7.45903677708983395971391842636, 8.946351793886263827401726241658, 9.469087472525089434434404833883, 10.72533771776725613164336504201, 11.22781726777200131033175772106
