| L(s) = 1 | + 2-s + 4-s − 7-s + 8-s − 2·13-s − 14-s + 16-s − 6·17-s − 4·19-s − 2·26-s − 28-s + 6·29-s − 4·31-s + 32-s − 6·34-s − 2·37-s − 4·38-s − 6·41-s − 8·43-s − 12·47-s + 49-s − 2·52-s + 6·53-s − 56-s + 6·58-s + 12·59-s + 2·61-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 0.554·13-s − 0.267·14-s + 1/4·16-s − 1.45·17-s − 0.917·19-s − 0.392·26-s − 0.188·28-s + 1.11·29-s − 0.718·31-s + 0.176·32-s − 1.02·34-s − 0.328·37-s − 0.648·38-s − 0.937·41-s − 1.21·43-s − 1.75·47-s + 1/7·49-s − 0.277·52-s + 0.824·53-s − 0.133·56-s + 0.787·58-s + 1.56·59-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−19.10191777534854, −18.07184693158180, −17.66688706812109, −16.79654499219246, −16.34416920040033, −15.55455869110539, −15.07323821372112, −14.48183596511669, −13.70888097661350, −13.03433170144647, −12.74791589908781, −11.71817186549171, −11.40476399462309, −10.33909737785192, −10.03830065066432, −8.845843570702888, −8.441852246728578, −7.278593507465635, −6.716380413533202, −6.114437678286766, −5.064861440359341, −4.495115166959840, −3.591307154225099, −2.653221769914648, −1.793002210675610, 0,
1.793002210675610, 2.653221769914648, 3.591307154225099, 4.495115166959840, 5.064861440359341, 6.114437678286766, 6.716380413533202, 7.278593507465635, 8.441852246728578, 8.845843570702888, 10.03830065066432, 10.33909737785192, 11.40476399462309, 11.71817186549171, 12.74791589908781, 13.03433170144647, 13.70888097661350, 14.48183596511669, 15.07323821372112, 15.55455869110539, 16.34416920040033, 16.79654499219246, 17.66688706812109, 18.07184693158180, 19.10191777534854
