| L(s) = 1 | + 7.15·7-s − 5.06i·11-s − 3.12·13-s − 8.72i·17-s + 20.1·19-s − 14.7i·23-s − 39.7i·29-s − 39.3·31-s + 34.8·37-s + 13.2i·41-s − 66.6·43-s + 16.9i·47-s + 2.18·49-s − 4.62i·53-s − 25.7i·59-s + ⋯ |
| L(s) = 1 | + 1.02·7-s − 0.460i·11-s − 0.240·13-s − 0.513i·17-s + 1.06·19-s − 0.640i·23-s − 1.37i·29-s − 1.27·31-s + 0.942·37-s + 0.323i·41-s − 1.54·43-s + 0.359i·47-s + 0.0445·49-s − 0.0873i·53-s − 0.436i·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.921364215\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.921364215\) |
| \(L(2)\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 - 7.15T + 49T^{2} \) |
| 11 | \( 1 + 5.06iT - 121T^{2} \) |
| 13 | \( 1 + 3.12T + 169T^{2} \) |
| 17 | \( 1 + 8.72iT - 289T^{2} \) |
| 19 | \( 1 - 20.1T + 361T^{2} \) |
| 23 | \( 1 + 14.7iT - 529T^{2} \) |
| 29 | \( 1 + 39.7iT - 841T^{2} \) |
| 31 | \( 1 + 39.3T + 961T^{2} \) |
| 37 | \( 1 - 34.8T + 1.36e3T^{2} \) |
| 41 | \( 1 - 13.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 66.6T + 1.84e3T^{2} \) |
| 47 | \( 1 - 16.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 4.62iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 25.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 12.1T + 3.72e3T^{2} \) |
| 67 | \( 1 - 106.T + 4.48e3T^{2} \) |
| 71 | \( 1 + 101. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 23.2T + 5.32e3T^{2} \) |
| 79 | \( 1 - 66.5T + 6.24e3T^{2} \) |
| 83 | \( 1 + 144. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 154. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 175.T + 9.40e3T^{2} \) |
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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
−8.260723280562742622240530280917, −7.88103050328408991041435031612, −7.03930337813547524506690537380, −6.12192158489592317707288306479, −5.25242456839145533489233722866, −4.66597600102954385446117159643, −3.64814655097272716157879103698, −2.64257916090716881586829861229, −1.61803943951132334362685406994, −0.45725820475203287175685390151,
1.21967825686906172900929715968, 2.00166268854074689736278714518, 3.22313996167539291993159642033, 4.11317844764321777561271563713, 5.12586768229404817305619616474, 5.47886570997282573262625847946, 6.72677493839129091568129794061, 7.39270923392085494875237203074, 8.043649702048571544167418046480, 8.803232395816917701407557956810
