L(s) = 1 | − 2·3-s − 2·7-s + 9-s + 5·13-s − 3·17-s − 2·19-s + 4·21-s − 6·23-s + 4·27-s + 3·29-s + 2·31-s − 7·37-s − 10·39-s + 3·41-s + 8·43-s − 6·47-s − 3·49-s + 6·51-s − 3·53-s + 4·57-s + 10·61-s − 2·63-s − 10·67-s + 12·69-s + 12·71-s − 14·73-s + 2·79-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 0.755·7-s + 1/3·9-s + 1.38·13-s − 0.727·17-s − 0.458·19-s + 0.872·21-s − 1.25·23-s + 0.769·27-s + 0.557·29-s + 0.359·31-s − 1.15·37-s − 1.60·39-s + 0.468·41-s + 1.21·43-s − 0.875·47-s − 3/7·49-s + 0.840·51-s − 0.412·53-s + 0.529·57-s + 1.28·61-s − 0.251·63-s − 1.22·67-s + 1.44·69-s + 1.42·71-s − 1.63·73-s + 0.225·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5075359973\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5075359973\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 18 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + 11 T + p T^{2} \) |
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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
−12.97705937721296, −12.59110975886633, −12.06634837111968, −11.75724438474453, −11.11943811302052, −10.80440095651926, −10.46921666582825, −9.879059956380801, −9.368907857610621, −8.828058696677816, −8.280273164888436, −7.996458655376876, −7.032687516965832, −6.661278651471409, −6.282181113921611, −5.899899616434919, −5.441561635244378, −4.769689334881846, −4.194996014700448, −3.776965874827062, −3.095971983191217, −2.472715029638035, −1.697418954985647, −1.026359801206637, −0.2487693461631769,
0.2487693461631769, 1.026359801206637, 1.697418954985647, 2.472715029638035, 3.095971983191217, 3.776965874827062, 4.194996014700448, 4.769689334881846, 5.441561635244378, 5.899899616434919, 6.282181113921611, 6.661278651471409, 7.032687516965832, 7.996458655376876, 8.280273164888436, 8.828058696677816, 9.368907857610621, 9.879059956380801, 10.46921666582825, 10.80440095651926, 11.11943811302052, 11.75724438474453, 12.06634837111968, 12.59110975886633, 12.97705937721296