L(s) = 1 | − 3-s + 3·7-s + 9-s + 4·11-s + 7·13-s − 4·17-s − 19-s − 3·21-s + 8·23-s − 27-s − 3·31-s − 4·33-s − 2·37-s − 7·39-s − 6·41-s + 11·43-s − 6·47-s + 2·49-s + 4·51-s − 6·53-s + 57-s − 6·59-s − 61-s + 3·63-s + 15·67-s − 8·69-s + 6·71-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1.13·7-s + 1/3·9-s + 1.20·11-s + 1.94·13-s − 0.970·17-s − 0.229·19-s − 0.654·21-s + 1.66·23-s − 0.192·27-s − 0.538·31-s − 0.696·33-s − 0.328·37-s − 1.12·39-s − 0.937·41-s + 1.67·43-s − 0.875·47-s + 2/7·49-s + 0.560·51-s − 0.824·53-s + 0.132·57-s − 0.781·59-s − 0.128·61-s + 0.377·63-s + 1.83·67-s − 0.963·69-s + 0.712·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.047461438\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.047461438\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 15 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−8.744089364880288473333648306528, −8.474427893688764358002948578751, −7.28123042860600156646586183211, −6.57667626296013395959479749509, −5.92670464741977493637424345453, −4.95097191993926904349126121511, −4.24622243695001846004341597039, −3.39303658339489081171608992934, −1.79557268711155458260765249071, −1.06145890089468242243699554760,
1.06145890089468242243699554760, 1.79557268711155458260765249071, 3.39303658339489081171608992934, 4.24622243695001846004341597039, 4.95097191993926904349126121511, 5.92670464741977493637424345453, 6.57667626296013395959479749509, 7.28123042860600156646586183211, 8.474427893688764358002948578751, 8.744089364880288473333648306528