| L(s) = 1 | + 2-s + 1.41·3-s + 4-s + 1.41·6-s − 7-s + 8-s + 1.00·9-s + 1.41·12-s − 1.41·13-s − 14-s + 16-s + 1.00·18-s − 1.41·19-s − 1.41·21-s + 1.41·24-s − 1.41·26-s − 28-s + 32-s + 1.00·36-s − 1.41·38-s − 2.00·39-s − 1.41·42-s + 1.41·48-s + 49-s − 1.41·52-s − 56-s − 2.00·57-s + ⋯ |
| L(s) = 1 | + 2-s + 1.41·3-s + 4-s + 1.41·6-s − 7-s + 8-s + 1.00·9-s + 1.41·12-s − 1.41·13-s − 14-s + 16-s + 1.00·18-s − 1.41·19-s − 1.41·21-s + 1.41·24-s − 1.41·26-s − 28-s + 32-s + 1.00·36-s − 1.41·38-s − 2.00·39-s − 1.41·42-s + 1.41·48-s + 49-s − 1.41·52-s − 56-s − 2.00·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.677120494\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.677120494\) |
| \(L(1)\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 - 1.41T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + 1.41T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + 1.41T + T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - 1.41T + T^{2} \) |
| 61 | \( 1 - 1.41T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + 1.41T + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 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
−9.832090625551487979729742969852, −8.913478514364769430029035608709, −8.076202761389319074767545581436, −7.21407151200468417538716643114, −6.61267764317405742780906068343, −5.53546840221130247644329135067, −4.40950238144763745485493038407, −3.65375259727954791325913932929, −2.72365004841739407044195635643, −2.15740706204371031633050772181,
2.15740706204371031633050772181, 2.72365004841739407044195635643, 3.65375259727954791325913932929, 4.40950238144763745485493038407, 5.53546840221130247644329135067, 6.61267764317405742780906068343, 7.21407151200468417538716643114, 8.076202761389319074767545581436, 8.913478514364769430029035608709, 9.832090625551487979729742969852
