| L(s) = 1 | + 2-s + 4-s + 7-s + 8-s − 9-s + 14-s + 16-s − 18-s − 2·23-s + 28-s + 32-s − 36-s − 2·46-s + 49-s + 56-s − 63-s + 64-s − 2·71-s − 72-s − 2·79-s + 81-s − 2·92-s + 98-s + 112-s + 2·113-s + ⋯ |
| L(s) = 1 | + 2-s + 4-s + 7-s + 8-s − 9-s + 14-s + 16-s − 18-s − 2·23-s + 28-s + 32-s − 36-s − 2·46-s + 49-s + 56-s − 63-s + 64-s − 2·71-s − 72-s − 2·79-s + 81-s − 2·92-s + 98-s + 112-s + 2·113-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
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.081859889\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.081859889\) |
| \(L(1)\) |
|
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 | \( 1 + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( ( 1 + T )^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 + T )^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 + T )^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 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
−10.00882980473781395286441698400, −8.725031599692830244359459696755, −8.037251577557102400555611085397, −7.32122865117550002331936504498, −6.14555230718548433550421110794, −5.62579229132141023663837039834, −4.69192565931160717702699352737, −3.87274410987393983486408048515, −2.73043243486413692739991743345, −1.75125247108869023879522052870,
1.75125247108869023879522052870, 2.73043243486413692739991743345, 3.87274410987393983486408048515, 4.69192565931160717702699352737, 5.62579229132141023663837039834, 6.14555230718548433550421110794, 7.32122865117550002331936504498, 8.037251577557102400555611085397, 8.725031599692830244359459696755, 10.00882980473781395286441698400
