L(s) = 1 | − 2·3-s − 2·4-s + 3·5-s + 7-s + 9-s + 4·12-s + 4·13-s − 6·15-s + 4·16-s + 3·17-s − 19-s − 6·20-s − 2·21-s + 4·25-s + 4·27-s − 2·28-s − 6·29-s − 4·31-s + 3·35-s − 2·36-s + 2·37-s − 8·39-s + 6·41-s + 43-s + 3·45-s − 3·47-s − 8·48-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 4-s + 1.34·5-s + 0.377·7-s + 1/3·9-s + 1.15·12-s + 1.10·13-s − 1.54·15-s + 16-s + 0.727·17-s − 0.229·19-s − 1.34·20-s − 0.436·21-s + 4/5·25-s + 0.769·27-s − 0.377·28-s − 1.11·29-s − 0.718·31-s + 0.507·35-s − 1/3·36-s + 0.328·37-s − 1.28·39-s + 0.937·41-s + 0.152·43-s + 0.447·45-s − 0.437·47-s − 1.15·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2299 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2299 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.244365176\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.244365176\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−9.100215838775092853341574624634, −8.428127489946401133860742721663, −7.42654336671663498602630753206, −6.22903281660809054167680244992, −5.78502585044634421931712450323, −5.30410557441328121366051727290, −4.42264240278191716580392851586, −3.36608228640882686115534658030, −1.83661141375158432347083933452, −0.807781280276512774607779040405,
0.807781280276512774607779040405, 1.83661141375158432347083933452, 3.36608228640882686115534658030, 4.42264240278191716580392851586, 5.30410557441328121366051727290, 5.78502585044634421931712450323, 6.22903281660809054167680244992, 7.42654336671663498602630753206, 8.428127489946401133860742721663, 9.100215838775092853341574624634