L(s) = 1 | − 0.199·2-s − 3-s − 1.96·4-s + 3.39·5-s + 0.199·6-s − 4.89·7-s + 0.788·8-s + 9-s − 0.675·10-s − 2.74·11-s + 1.96·12-s − 0.0240·13-s + 0.975·14-s − 3.39·15-s + 3.76·16-s − 6.00·17-s − 0.199·18-s − 7.22·19-s − 6.65·20-s + 4.89·21-s + 0.545·22-s + 3.00·23-s − 0.788·24-s + 6.51·25-s + 0.00478·26-s − 27-s + 9.59·28-s + ⋯ |
L(s) = 1 | − 0.140·2-s − 0.577·3-s − 0.980·4-s + 1.51·5-s + 0.0812·6-s − 1.85·7-s + 0.278·8-s + 0.333·9-s − 0.213·10-s − 0.826·11-s + 0.565·12-s − 0.00666·13-s + 0.260·14-s − 0.876·15-s + 0.940·16-s − 1.45·17-s − 0.0469·18-s − 1.65·19-s − 1.48·20-s + 1.06·21-s + 0.116·22-s + 0.627·23-s − 0.160·24-s + 1.30·25-s + 0.000938·26-s − 0.192·27-s + 1.81·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 103 | \( 1 + T \) |
good | 2 | \( 1 + 0.199T + 2T^{2} \) |
| 5 | \( 1 - 3.39T + 5T^{2} \) |
| 7 | \( 1 + 4.89T + 7T^{2} \) |
| 11 | \( 1 + 2.74T + 11T^{2} \) |
| 13 | \( 1 + 0.0240T + 13T^{2} \) |
| 17 | \( 1 + 6.00T + 17T^{2} \) |
| 19 | \( 1 + 7.22T + 19T^{2} \) |
| 23 | \( 1 - 3.00T + 23T^{2} \) |
| 29 | \( 1 + 7.29T + 29T^{2} \) |
| 31 | \( 1 + 6.65T + 31T^{2} \) |
| 37 | \( 1 - 1.94T + 37T^{2} \) |
| 41 | \( 1 + 3.76T + 41T^{2} \) |
| 43 | \( 1 - 10.3T + 43T^{2} \) |
| 47 | \( 1 - 5.72T + 47T^{2} \) |
| 53 | \( 1 - 2.58T + 53T^{2} \) |
| 59 | \( 1 - 9.00T + 59T^{2} \) |
| 61 | \( 1 - 1.30T + 61T^{2} \) |
| 67 | \( 1 - 6.05T + 67T^{2} \) |
| 71 | \( 1 + 11.0T + 71T^{2} \) |
| 73 | \( 1 + 4.93T + 73T^{2} \) |
| 79 | \( 1 + 5.86T + 79T^{2} \) |
| 83 | \( 1 - 1.16T + 83T^{2} \) |
| 89 | \( 1 - 1.65T + 89T^{2} \) |
| 97 | \( 1 - 5.34T + 97T^{2} \) |
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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
−10.78524883511775702473262744120, −10.24829730667054905198939264789, −9.329063806757883109501850121795, −8.904724885736600666677570516109, −7.05690371110631180166736716921, −6.11562895202761770152224626632, −5.43752702566080964602349111560, −4.05544324208852895315881496275, −2.38545319789238017269990547835, 0,
2.38545319789238017269990547835, 4.05544324208852895315881496275, 5.43752702566080964602349111560, 6.11562895202761770152224626632, 7.05690371110631180166736716921, 8.904724885736600666677570516109, 9.329063806757883109501850121795, 10.24829730667054905198939264789, 10.78524883511775702473262744120