| 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
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.295024346\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.295024346\) |
| \(L(1)\) |
|
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
−10.10337105892282530438826706684, −9.128431405420742852408896083804, −7.80062367076982391498526349108, −6.87119116955904076246851068767, −6.20722443103545889364585251590, −5.73041708354120256504712037816, −4.89543848695057668836629153368, −3.82834728250147071235564423871, −3.00160949841360440982666450832, −1.24202895758901682203803852052,
1.24202895758901682203803852052, 3.00160949841360440982666450832, 3.82834728250147071235564423871, 4.89543848695057668836629153368, 5.73041708354120256504712037816, 6.20722443103545889364585251590, 6.87119116955904076246851068767, 7.80062367076982391498526349108, 9.128431405420742852408896083804, 10.10337105892282530438826706684
