| L(s) = 1 | + i·3-s + 7-s − 9-s + 6i·11-s − 2i·13-s − 4i·19-s + i·21-s − 6·23-s + 5·25-s − i·27-s − 6i·29-s − 8·31-s − 6·33-s + 2i·37-s + 2·39-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 0.377·7-s − 0.333·9-s + 1.80i·11-s − 0.554i·13-s − 0.917i·19-s + 0.218i·21-s − 1.25·23-s + 25-s − 0.192i·27-s − 1.11i·29-s − 1.43·31-s − 1.04·33-s + 0.328i·37-s + 0.320·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 - 5T^{2} \) |
| 11 | \( 1 - 6iT - 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 12T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 12T + 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 - 10iT - 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 + 12T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−7.983081919844388091019770191939, −7.16871189149650957205261704801, −6.60145928989985901456930619813, −5.54229606706781697400865913072, −4.87548881225291001669457992762, −4.37511585051108608285918087555, −3.46445598051237037944234230449, −2.44472394532890852593298668173, −1.64127100441604991309627763858, 0,
1.29227534929850042669879402250, 2.06875207797482378266165752435, 3.32011788639021068020984106117, 3.72163334974701637072978281481, 5.02742272292131235884624338243, 5.57844285564960434267849504143, 6.39204791689497160412699890303, 6.89595211369618080201968592524, 7.942696774340037832322275123907
