| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 2·7-s + 8-s + 9-s + 12-s − 2·13-s + 2·14-s + 16-s + 2·17-s + 18-s + 2·21-s + 2·23-s + 24-s − 2·26-s + 27-s + 2·28-s − 4·29-s − 4·31-s + 32-s + 2·34-s + 36-s − 2·37-s − 2·39-s + 4·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.288·12-s − 0.554·13-s + 0.534·14-s + 1/4·16-s + 0.485·17-s + 0.235·18-s + 0.436·21-s + 0.417·23-s + 0.204·24-s − 0.392·26-s + 0.192·27-s + 0.377·28-s − 0.742·29-s − 0.718·31-s + 0.176·32-s + 0.342·34-s + 1/6·36-s − 0.328·37-s − 0.320·39-s + 0.624·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−14.75654030317799, −14.24148579321097, −13.70313035404209, −13.23404333659104, −12.70252001792855, −12.24363347567322, −11.68420173454987, −11.14841607078408, −10.73308867778704, −10.02876775257974, −9.544180372095051, −8.979528030702262, −8.300282538313269, −7.808480684733375, −7.405573196624030, −6.775162813735085, −6.182803358442343, −5.375327421858632, −5.017193826333262, −4.457203892603525, −3.726270140343933, −3.218355540779991, −2.561747488505352, −1.785816139200583, −1.344212059432826, 0,
1.344212059432826, 1.785816139200583, 2.561747488505352, 3.218355540779991, 3.726270140343933, 4.457203892603525, 5.017193826333262, 5.375327421858632, 6.182803358442343, 6.775162813735085, 7.405573196624030, 7.808480684733375, 8.300282538313269, 8.979528030702262, 9.544180372095051, 10.02876775257974, 10.73308867778704, 11.14841607078408, 11.68420173454987, 12.24363347567322, 12.70252001792855, 13.23404333659104, 13.70313035404209, 14.24148579321097, 14.75654030317799
