| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 5·7-s − 8-s + 9-s − 11-s − 12-s − 5·14-s + 16-s − 4·17-s − 18-s + 3·19-s − 5·21-s + 22-s − 23-s + 24-s − 27-s + 5·28-s + 6·29-s − 31-s − 32-s + 33-s + 4·34-s + 36-s − 4·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 1.88·7-s − 0.353·8-s + 1/3·9-s − 0.301·11-s − 0.288·12-s − 1.33·14-s + 1/4·16-s − 0.970·17-s − 0.235·18-s + 0.688·19-s − 1.09·21-s + 0.213·22-s − 0.208·23-s + 0.204·24-s − 0.192·27-s + 0.944·28-s + 1.11·29-s − 0.179·31-s − 0.176·32-s + 0.174·33-s + 0.685·34-s + 1/6·36-s − 0.657·37-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
| \(L(1)\) |
\(\approx\) |
\(1.401781246\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.401781246\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 5 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 15 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−8.324144007286194476239608201124, −7.65953445349591187460531797012, −7.05127744977499064075254649470, −6.19342098565817811456718165780, −5.26005152525608288771498772826, −4.82598050501665912095185675835, −3.91147502021945010746086829763, −2.51938515684160805822243348837, −1.72974318761389879716166327223, −0.78630454339232826724681158124,
0.78630454339232826724681158124, 1.72974318761389879716166327223, 2.51938515684160805822243348837, 3.91147502021945010746086829763, 4.82598050501665912095185675835, 5.26005152525608288771498772826, 6.19342098565817811456718165780, 7.05127744977499064075254649470, 7.65953445349591187460531797012, 8.324144007286194476239608201124
