| L(s) = 1 | + (1.64 + 0.541i)3-s + 3.29i·7-s + (2.41 + 1.78i)9-s + 2.51i·11-s − 4.65i·13-s + 3.69i·17-s + 0.828·19-s + (−1.78 + 5.41i)21-s + 2.61·23-s + (3.00 + 4.23i)27-s − 6.08·29-s + 1.17i·31-s + (−1.36 + 4.14i)33-s + 1.92i·37-s + (2.51 − 7.65i)39-s + ⋯ |
| L(s) = 1 | + (0.949 + 0.312i)3-s + 1.24i·7-s + (0.804 + 0.593i)9-s + 0.759i·11-s − 1.29i·13-s + 0.896i·17-s + 0.190·19-s + (−0.388 + 1.18i)21-s + 0.544·23-s + (0.578 + 0.815i)27-s − 1.12·29-s + 0.210i·31-s + (−0.237 + 0.721i)33-s + 0.316i·37-s + (0.403 − 1.22i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0748 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0748 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.431200751\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.431200751\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.64 - 0.541i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 3.29iT - 7T^{2} \) |
| 11 | \( 1 - 2.51iT - 11T^{2} \) |
| 13 | \( 1 + 4.65iT - 13T^{2} \) |
| 17 | \( 1 - 3.69iT - 17T^{2} \) |
| 19 | \( 1 - 0.828T + 19T^{2} \) |
| 23 | \( 1 - 2.61T + 23T^{2} \) |
| 29 | \( 1 + 6.08T + 29T^{2} \) |
| 31 | \( 1 - 1.17iT - 31T^{2} \) |
| 37 | \( 1 - 1.92iT - 37T^{2} \) |
| 41 | \( 1 - 8.59iT - 41T^{2} \) |
| 43 | \( 1 - 6.01T + 43T^{2} \) |
| 47 | \( 1 - 2.61T + 47T^{2} \) |
| 53 | \( 1 + 4.59T + 53T^{2} \) |
| 59 | \( 1 - 2.51iT - 59T^{2} \) |
| 61 | \( 1 + 8.48iT - 61T^{2} \) |
| 67 | \( 1 - 3.29T + 67T^{2} \) |
| 71 | \( 1 + 7.12T + 71T^{2} \) |
| 73 | \( 1 + 6.58T + 73T^{2} \) |
| 79 | \( 1 - 16.4iT - 79T^{2} \) |
| 83 | \( 1 + 9.37iT - 83T^{2} \) |
| 89 | \( 1 - 5.03iT - 89T^{2} \) |
| 97 | \( 1 + 2.72T + 97T^{2} \) |
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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
−9.195658509230184100439069669613, −8.357000987378902576086501304404, −7.893684547691721042447954125184, −7.02079497973784093590823073098, −5.90207506640837674105128800053, −5.22892436454677983318711073779, −4.30497867722526795864081032818, −3.25643489946824470237989922559, −2.57380885990890275108768449142, −1.58634001802288485298772378580,
0.72895384355013534130811304516, 1.86166047435733258249673915997, 2.99031737464088193167992361008, 3.87340249036856369896574215990, 4.44658809688501341368223112040, 5.69871560023755987365564765722, 6.81254454136628260068100362327, 7.24435740979385179817958290794, 7.85564643456811809259267935939, 8.990986917204953811579000799234
