L(s) = 1 | − 1.24·2-s + 1.80·3-s + 0.554·4-s − 2.24·6-s + 0.554·8-s + 2.24·9-s + 0.999·12-s − 1.24·16-s − 2.80·18-s + 1.24·19-s + 1.00·24-s + 2.24·27-s − 1.80·29-s + 0.999·32-s + 1.24·36-s − 1.24·37-s − 1.55·38-s + 0.445·43-s − 2.24·48-s + 49-s − 2.80·54-s + 2.24·57-s + 2.24·58-s + 71-s + 1.24·72-s + 0.445·73-s + 1.55·74-s + ⋯ |
L(s) = 1 | − 1.24·2-s + 1.80·3-s + 0.554·4-s − 2.24·6-s + 0.554·8-s + 2.24·9-s + 0.999·12-s − 1.24·16-s − 2.80·18-s + 1.24·19-s + 1.00·24-s + 2.24·27-s − 1.80·29-s + 0.999·32-s + 1.24·36-s − 1.24·37-s − 1.55·38-s + 0.445·43-s − 2.24·48-s + 49-s − 2.80·54-s + 2.24·57-s + 2.24·58-s + 71-s + 1.24·72-s + 0.445·73-s + 1.55·74-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.134288806\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.134288806\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 71 | \( 1 - T \) |
good | 2 | \( 1 + 1.24T + T^{2} \) |
| 3 | \( 1 - 1.80T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - 1.24T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + 1.80T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + 1.24T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - 0.445T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 73 | \( 1 - 0.445T + T^{2} \) |
| 79 | \( 1 + 0.445T + T^{2} \) |
| 83 | \( 1 + 1.24T + T^{2} \) |
| 89 | \( 1 + 1.80T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.330132796341729703608864420600, −8.788800985474931443322493569254, −8.092999675564426862480072422022, −7.44634167506164238076235736069, −6.98849590131557118033005282891, −5.40012912993915675875647282975, −4.18730425919183222403248140492, −3.37656538460639035528715631921, −2.30469465717266151622483926916, −1.39693348684821972064541686038,
1.39693348684821972064541686038, 2.30469465717266151622483926916, 3.37656538460639035528715631921, 4.18730425919183222403248140492, 5.40012912993915675875647282975, 6.98849590131557118033005282891, 7.44634167506164238076235736069, 8.092999675564426862480072422022, 8.788800985474931443322493569254, 9.330132796341729703608864420600