| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 3·7-s + 8-s + 9-s − 3·11-s + 12-s − 13-s + 3·14-s + 16-s − 2·17-s + 18-s + 3·21-s − 3·22-s + 4·23-s + 24-s − 26-s + 27-s + 3·28-s + 10·29-s + 31-s + 32-s − 3·33-s − 2·34-s + 36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 1.13·7-s + 0.353·8-s + 1/3·9-s − 0.904·11-s + 0.288·12-s − 0.277·13-s + 0.801·14-s + 1/4·16-s − 0.485·17-s + 0.235·18-s + 0.654·21-s − 0.639·22-s + 0.834·23-s + 0.204·24-s − 0.196·26-s + 0.192·27-s + 0.566·28-s + 1.85·29-s + 0.179·31-s + 0.176·32-s − 0.522·33-s − 0.342·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.426220129\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.426220129\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−8.223510930822521760547925176185, −7.62833152241178359033636775670, −6.91439807521218786009300843759, −6.07664231388497226717195123261, −4.93775376934719768962601008975, −4.84738744072462948184704587222, −3.83344607709084577830587823370, −2.77001574745193921936396912607, −2.26783051060877405598204079151, −1.07793221925619497694495369479,
1.07793221925619497694495369479, 2.26783051060877405598204079151, 2.77001574745193921936396912607, 3.83344607709084577830587823370, 4.84738744072462948184704587222, 4.93775376934719768962601008975, 6.07664231388497226717195123261, 6.91439807521218786009300843759, 7.62833152241178359033636775670, 8.223510930822521760547925176185
