L(s) = 1 | − 2·3-s + 9-s − 2·17-s − 2·19-s + 8·23-s + 4·27-s + 2·29-s + 4·31-s + 6·37-s + 2·41-s + 8·43-s + 4·47-s + 4·51-s + 10·53-s + 4·57-s + 6·59-s − 4·61-s − 12·67-s − 16·69-s − 14·73-s + 8·79-s − 11·81-s − 6·83-s − 4·87-s − 10·89-s − 8·93-s − 2·97-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 1/3·9-s − 0.485·17-s − 0.458·19-s + 1.66·23-s + 0.769·27-s + 0.371·29-s + 0.718·31-s + 0.986·37-s + 0.312·41-s + 1.21·43-s + 0.583·47-s + 0.560·51-s + 1.37·53-s + 0.529·57-s + 0.781·59-s − 0.512·61-s − 1.46·67-s − 1.92·69-s − 1.63·73-s + 0.900·79-s − 1.22·81-s − 0.658·83-s − 0.428·87-s − 1.05·89-s − 0.829·93-s − 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.281955822\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.281955822\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 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
−15.82405765005163, −15.09693119252884, −14.79230825543658, −14.00675929628737, −13.36900657864091, −12.91073723587234, −12.28972355809745, −11.81498146814740, −11.21878183900450, −10.80610808729591, −10.36002821240915, −9.604735318591604, −8.882095097205039, −8.518256119484362, −7.563653662121162, −7.020168058158480, −6.436078599670725, −5.843726739378018, −5.345202499043781, −4.539733454781880, −4.204446779708564, −2.999772450411071, −2.482486436164423, −1.247481123266087, −0.5635329325736111,
0.5635329325736111, 1.247481123266087, 2.482486436164423, 2.999772450411071, 4.204446779708564, 4.539733454781880, 5.345202499043781, 5.843726739378018, 6.436078599670725, 7.020168058158480, 7.563653662121162, 8.518256119484362, 8.882095097205039, 9.604735318591604, 10.36002821240915, 10.80610808729591, 11.21878183900450, 11.81498146814740, 12.28972355809745, 12.91073723587234, 13.36900657864091, 14.00675929628737, 14.79230825543658, 15.09693119252884, 15.82405765005163