| L(s) = 1 | + 3-s + 3·5-s + 7-s − 2·9-s + 3·11-s + 13-s + 3·15-s − 3·19-s + 21-s + 23-s + 4·25-s − 5·27-s + 2·29-s − 10·31-s + 3·33-s + 3·35-s − 2·37-s + 39-s + 2·41-s + 8·43-s − 6·45-s − 6·47-s + 49-s + 8·53-s + 9·55-s − 3·57-s − 6·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.34·5-s + 0.377·7-s − 2/3·9-s + 0.904·11-s + 0.277·13-s + 0.774·15-s − 0.688·19-s + 0.218·21-s + 0.208·23-s + 4/5·25-s − 0.962·27-s + 0.371·29-s − 1.79·31-s + 0.522·33-s + 0.507·35-s − 0.328·37-s + 0.160·39-s + 0.312·41-s + 1.21·43-s − 0.894·45-s − 0.875·47-s + 1/7·49-s + 1.09·53-s + 1.21·55-s − 0.397·57-s − 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 133952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−13.91596616230820, −13.34764884906275, −12.78183073354360, −12.40542666971824, −11.73863351967268, −11.20368934663234, −10.80045404208188, −10.34425664002925, −9.628572306300642, −9.289940870191044, −8.859009207439847, −8.575409485772268, −7.850545544421224, −7.321398854911230, −6.707808766377434, −6.151140811432945, −5.745391424514564, −5.363161615294989, −4.554850819640975, −4.043807338874830, −3.356154298228264, −2.813218990402364, −2.107952392104159, −1.755382901514892, −1.100247112135924, 0,
1.100247112135924, 1.755382901514892, 2.107952392104159, 2.813218990402364, 3.356154298228264, 4.043807338874830, 4.554850819640975, 5.363161615294989, 5.745391424514564, 6.151140811432945, 6.707808766377434, 7.321398854911230, 7.850545544421224, 8.575409485772268, 8.859009207439847, 9.289940870191044, 9.628572306300642, 10.34425664002925, 10.80045404208188, 11.20368934663234, 11.73863351967268, 12.40542666971824, 12.78183073354360, 13.34764884906275, 13.91596616230820
