| L(s) = 1 | − 2-s + 4-s − 5-s − 8-s + 9-s + 10-s − 11-s + 13-s + 16-s − 18-s − 19-s − 20-s + 22-s + 23-s + 25-s − 26-s − 32-s + 36-s + 37-s + 38-s + 40-s − 41-s − 44-s − 45-s − 46-s + 47-s − 50-s + ⋯ |
| L(s) = 1 | − 2-s + 4-s − 5-s − 8-s + 9-s + 10-s − 11-s + 13-s + 16-s − 18-s − 19-s − 20-s + 22-s + 23-s + 25-s − 26-s − 32-s + 36-s + 37-s + 38-s + 40-s − 41-s − 44-s − 45-s − 46-s + 47-s − 50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6488982764\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6488982764\) |
| \(L(1)\) |
|
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 | \( 1 \) |
| good | 3 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( 1 - T + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( 1 + T + T^{2} \) |
| 23 | \( 1 - T + T^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( 1 - T + T^{2} \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( 1 - T + T^{2} \) |
| 53 | \( 1 - T + T^{2} \) |
| 59 | \( ( 1 - T )^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 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
−9.229791102204469147103232179163, −8.487146402096743553080070494077, −7.957625375746181345227001572003, −7.15539334433045220504655879093, −6.60197150286083182339309397712, −5.46239217716715851815642871756, −4.33274959514248666790607144566, −3.43747193100768837314523766124, −2.32007427042724961223281344873, −0.944624418819696518537810846011,
0.944624418819696518537810846011, 2.32007427042724961223281344873, 3.43747193100768837314523766124, 4.33274959514248666790607144566, 5.46239217716715851815642871756, 6.60197150286083182339309397712, 7.15539334433045220504655879093, 7.957625375746181345227001572003, 8.487146402096743553080070494077, 9.229791102204469147103232179163
