L(s) = 1 | − 2-s + 4-s − 5-s + 7-s − 8-s + 9-s + 10-s + 13-s − 14-s + 16-s − 18-s − 20-s + 25-s − 26-s + 28-s − 32-s − 35-s + 36-s + 40-s − 45-s + 49-s − 50-s + 52-s − 56-s − 2·61-s + 63-s + 64-s + ⋯ |
L(s) = 1 | − 2-s + 4-s − 5-s + 7-s − 8-s + 9-s + 10-s + 13-s − 14-s + 16-s − 18-s − 20-s + 25-s − 26-s + 28-s − 32-s − 35-s + 36-s + 40-s − 45-s + 49-s − 50-s + 52-s − 56-s − 2·61-s + 63-s + 64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8930200389\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8930200389\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
good | 3 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( 1 + T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 + T )^{2} \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 + T )^{2} \) |
| 83 | \( ( 1 - T )^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−8.656231741704359658911783055084, −7.950257323547277274464784444332, −7.51162952057775941231783369642, −6.80108717976883725749730819390, −5.93974588358676565257066301307, −4.82972923479146520731236359615, −4.05012971685181710953627697113, −3.17302041835627947402778284233, −1.87331756370275506192083429272, −1.02265494596725043487155337052,
1.02265494596725043487155337052, 1.87331756370275506192083429272, 3.17302041835627947402778284233, 4.05012971685181710953627697113, 4.82972923479146520731236359615, 5.93974588358676565257066301307, 6.80108717976883725749730819390, 7.51162952057775941231783369642, 7.950257323547277274464784444332, 8.656231741704359658911783055084