L(s) = 1 | − 4-s + 5-s − 9-s − 2·11-s + 16-s + 19-s − 20-s + 25-s + 36-s + 2·44-s − 45-s + 49-s − 2·55-s − 2·61-s − 64-s − 76-s + 80-s + 81-s + 95-s + 2·99-s − 100-s − 2·101-s + ⋯ |
L(s) = 1 | − 4-s + 5-s − 9-s − 2·11-s + 16-s + 19-s − 20-s + 25-s + 36-s + 2·44-s − 45-s + 49-s − 2·55-s − 2·61-s − 64-s − 76-s + 80-s + 81-s + 95-s + 2·99-s − 100-s − 2·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4985172696\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4985172696\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 19 | \( 1 - T \) |
good | 2 | \( 1 + T^{2} \) |
| 3 | \( 1 + T^{2} \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( ( 1 + T )^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 + T )^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 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
−13.90530768021683048703931500792, −13.47265417168699841911242379081, −12.42493324594992440640465564801, −10.80601861381863672990663374102, −9.891081373460128606623069373424, −8.844029451101821452107687940364, −7.76012765378292796390012623669, −5.76418806130242862195870775024, −5.05764801474219645507661583018, −2.87350641693744207631912289409,
2.87350641693744207631912289409, 5.05764801474219645507661583018, 5.76418806130242862195870775024, 7.76012765378292796390012623669, 8.844029451101821452107687940364, 9.891081373460128606623069373424, 10.80601861381863672990663374102, 12.42493324594992440640465564801, 13.47265417168699841911242379081, 13.90530768021683048703931500792