| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 7-s − 8-s + 9-s + 12-s − 2·13-s + 14-s + 16-s − 18-s − 2·19-s − 21-s − 24-s − 5·25-s + 2·26-s + 27-s − 28-s + 6·29-s + 2·31-s − 32-s + 36-s + 2·37-s + 2·38-s − 2·39-s + 42-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.288·12-s − 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.235·18-s − 0.458·19-s − 0.218·21-s − 0.204·24-s − 25-s + 0.392·26-s + 0.192·27-s − 0.188·28-s + 1.11·29-s + 0.359·31-s − 0.176·32-s + 1/6·36-s + 0.328·37-s + 0.324·38-s − 0.320·39-s + 0.154·42-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5082 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5082 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−7.917511782936980017432170305194, −7.36329706847033770459443227642, −6.54681723608152413257938391600, −5.95126526016212345113256401478, −4.84928262310734676871752984376, −4.03977724610078358039750840478, −3.04842425522471748432463102719, −2.39295039746984162234114406993, −1.36782001529104331538153017599, 0,
1.36782001529104331538153017599, 2.39295039746984162234114406993, 3.04842425522471748432463102719, 4.03977724610078358039750840478, 4.84928262310734676871752984376, 5.95126526016212345113256401478, 6.54681723608152413257938391600, 7.36329706847033770459443227642, 7.917511782936980017432170305194
