| L(s) = 1 | + 1.41·2-s + 1.00·4-s − i·7-s + (0.707 + 0.707i)11-s − 1.41i·14-s − 0.999·16-s + (1.00 + 1.00i)22-s + 1.41i·23-s − 25-s − 1.00i·28-s − 1.41·29-s − 1.41·32-s − 2i·43-s + (0.707 + 0.707i)44-s + 2.00i·46-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.00·4-s − i·7-s + (0.707 + 0.707i)11-s − 1.41i·14-s − 0.999·16-s + (1.00 + 1.00i)22-s + 1.41i·23-s − 25-s − 1.00i·28-s − 1.41·29-s − 1.41·32-s − 2i·43-s + (0.707 + 0.707i)44-s + 2.00i·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.807502548\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.807502548\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 + (-0.707 - 0.707i)T \) |
| good | 2 | \( 1 - 1.41T + T^{2} \) |
| 5 | \( 1 + T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 - 1.41iT - T^{2} \) |
| 29 | \( 1 + 1.41T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 2iT - T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 - 1.41iT - T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 - 2T + T^{2} \) |
| 71 | \( 1 + 1.41iT - T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−10.94459366472756590110234086143, −9.836336853828360550811599267837, −9.105044716669220011842860157970, −7.62722574924534401548871332073, −7.00668484461153593923150670846, −5.98679320596978415089870214585, −5.09223585859111139745386502447, −4.02400369787465170299237405787, −3.57419044443966412427269287338, −1.91950353999548897104714992072,
2.20328862225480509068748782173, 3.29845231913263561702751950053, 4.23419515357445991036304932832, 5.28836838029265942518065760297, 6.02727282156005852611545332929, 6.69492256948612319727346345664, 8.124457640821783930041700643133, 8.961978022691072382958075945634, 9.793287747494193724534950925800, 11.31470028319016370870012503410
