L(s) = 1 | − 3-s + 5·7-s + 9-s + 5·11-s − 13-s + 3·17-s − 4·19-s − 5·21-s − 5·23-s − 27-s − 4·29-s − 5·33-s + 7·37-s + 39-s + 11·41-s + 12·43-s − 6·47-s + 18·49-s − 3·51-s − 53-s + 4·57-s + 12·59-s − 7·61-s + 5·63-s − 4·67-s + 5·69-s − 7·71-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1.88·7-s + 1/3·9-s + 1.50·11-s − 0.277·13-s + 0.727·17-s − 0.917·19-s − 1.09·21-s − 1.04·23-s − 0.192·27-s − 0.742·29-s − 0.870·33-s + 1.15·37-s + 0.160·39-s + 1.71·41-s + 1.82·43-s − 0.875·47-s + 18/7·49-s − 0.420·51-s − 0.137·53-s + 0.529·57-s + 1.56·59-s − 0.896·61-s + 0.629·63-s − 0.488·67-s + 0.601·69-s − 0.830·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.499889755\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.499889755\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 - 5 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 11 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−7.73940748831326276431211845141, −7.34755893728189458798254328387, −6.27718295737360732431148322724, −5.84098912679039730444238635550, −5.02410032482078224271454267736, −4.20269918786637403894536926895, −3.99610724676521574372818103553, −2.41899683268664310364411258353, −1.63881857837667446731162069344, −0.886235258664121587032391820755,
0.886235258664121587032391820755, 1.63881857837667446731162069344, 2.41899683268664310364411258353, 3.99610724676521574372818103553, 4.20269918786637403894536926895, 5.02410032482078224271454267736, 5.84098912679039730444238635550, 6.27718295737360732431148322724, 7.34755893728189458798254328387, 7.73940748831326276431211845141