| L(s) = 1 | + (0.311 − 2.21i)5-s + i·7-s − 3.80·11-s + 0.622i·13-s − 4.42i·17-s + 0.622·19-s + 2.62i·23-s + (−4.80 − 1.37i)25-s + 9.61·29-s + 0.622·31-s + (2.21 + 0.311i)35-s + 1.24i·37-s − 4.62·41-s + 4.85i·43-s − 11.6i·47-s + ⋯ |
| L(s) = 1 | + (0.139 − 0.990i)5-s + 0.377i·7-s − 1.14·11-s + 0.172i·13-s − 1.07i·17-s + 0.142·19-s + 0.546i·23-s + (−0.961 − 0.275i)25-s + 1.78·29-s + 0.111·31-s + (0.374 + 0.0525i)35-s + 0.204i·37-s − 0.721·41-s + 0.740i·43-s − 1.69i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.139i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 - 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3665626350\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3665626350\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.311 + 2.21i)T \) |
| 7 | \( 1 - iT \) |
| good | 11 | \( 1 + 3.80T + 11T^{2} \) |
| 13 | \( 1 - 0.622iT - 13T^{2} \) |
| 17 | \( 1 + 4.42iT - 17T^{2} \) |
| 19 | \( 1 - 0.622T + 19T^{2} \) |
| 23 | \( 1 - 2.62iT - 23T^{2} \) |
| 29 | \( 1 - 9.61T + 29T^{2} \) |
| 31 | \( 1 - 0.622T + 31T^{2} \) |
| 37 | \( 1 - 1.24iT - 37T^{2} \) |
| 41 | \( 1 + 4.62T + 41T^{2} \) |
| 43 | \( 1 - 4.85iT - 43T^{2} \) |
| 47 | \( 1 + 11.6iT - 47T^{2} \) |
| 53 | \( 1 + 13.4iT - 53T^{2} \) |
| 59 | \( 1 + 11.6T + 59T^{2} \) |
| 61 | \( 1 + 8.10T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 - 2.56T + 71T^{2} \) |
| 73 | \( 1 - 10.9iT - 73T^{2} \) |
| 79 | \( 1 + 6.75T + 79T^{2} \) |
| 83 | \( 1 - 11.6iT - 83T^{2} \) |
| 89 | \( 1 + 8.23T + 89T^{2} \) |
| 97 | \( 1 + 4.23iT - 97T^{2} \) |
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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
−8.143733587911048864615378531831, −7.17042989504385810859123607442, −6.41247221394635341019152079457, −5.35777291431187654150065423724, −5.12407152739938568635995132709, −4.31939803190949385535001845913, −3.15302763512907222261324587447, −2.39406987055977735011620008069, −1.29719185731468485723872521634, −0.097133414712482843201546568408,
1.45473302498275342756281985128, 2.63811689677487952242117107850, 3.10076145798553163204916222820, 4.16803674188944107692167173230, 4.89735858400847051250647929374, 5.95400376383535770965229400186, 6.34532190665231025936933609787, 7.26112648505085179366602685893, 7.82221682658568990223223354809, 8.428807316314718812179528861322
