L(s) = 1 | − i·3-s + 2i·5-s − 7-s − 9-s − 4i·11-s − 6i·13-s + 2·15-s − 2·17-s − 4i·19-s + i·21-s − 4·23-s + 25-s + i·27-s − 2i·29-s − 8·31-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + 0.894i·5-s − 0.377·7-s − 0.333·9-s − 1.20i·11-s − 1.66i·13-s + 0.516·15-s − 0.485·17-s − 0.917i·19-s + 0.218i·21-s − 0.834·23-s + 0.200·25-s + 0.192i·27-s − 0.371i·29-s − 1.43·31-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 | \( 1 - 2iT - 5T^{2} \) |
| 11 | \( 1 + 4iT - 11T^{2} \) |
| 13 | \( 1 + 6iT - 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 2iT - 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 - 10iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 10iT - 53T^{2} \) |
| 59 | \( 1 + 12iT - 59T^{2} \) |
| 61 | \( 1 - 10iT - 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−7.72213915376762324295688574673, −7.00005604742938908064987684111, −6.26426247803484509552425333969, −5.83618228497125908807436319511, −4.95201737198047055716805399331, −3.71516227340279091913054824687, −2.98805194554284973854058449427, −2.54705289132506810292487891514, −1.07402376152324713741471343362, 0,
1.67371036811037920712114210785, 2.25480581246777942234856344863, 3.88383371842442626249107445350, 4.01473040705127168586474644964, 4.96385823607857016413063572385, 5.54797915635949854005537895631, 6.55197057222954709846150037150, 7.12848448210342444478720069643, 7.980429295829972763339150929376