| L(s) = 1 | + 3-s − 2·7-s + 9-s + 6·13-s − 3·17-s − 2·19-s − 2·21-s − 3·23-s + 27-s − 4·29-s + 31-s − 10·37-s + 6·39-s + 3·41-s − 43-s − 7·47-s − 3·49-s − 3·51-s − 9·53-s − 2·57-s − 3·59-s − 2·61-s − 2·63-s + 11·67-s − 3·69-s − 14·71-s + 3·79-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.755·7-s + 1/3·9-s + 1.66·13-s − 0.727·17-s − 0.458·19-s − 0.436·21-s − 0.625·23-s + 0.192·27-s − 0.742·29-s + 0.179·31-s − 1.64·37-s + 0.960·39-s + 0.468·41-s − 0.152·43-s − 1.02·47-s − 3/7·49-s − 0.420·51-s − 1.23·53-s − 0.264·57-s − 0.390·59-s − 0.256·61-s − 0.251·63-s + 1.34·67-s − 0.361·69-s − 1.66·71-s + 0.337·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.933546039\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.933546039\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−14.45321808331799, −13.85919477169545, −13.47194918693201, −13.04802280634032, −12.62992701412029, −11.96588218186651, −11.34441830753419, −10.79374361084678, −10.43909706200871, −9.711530581510352, −9.200692640599060, −8.835242488919892, −8.148468477808775, −7.889501076552205, −6.867988284617111, −6.606379151713716, −6.050192395724108, −5.427981269491059, −4.566682908108977, −4.003463721891949, −3.403200859788131, −3.035457250080344, −1.981200672015323, −1.598725618926986, −0.4531888134947494,
0.4531888134947494, 1.598725618926986, 1.981200672015323, 3.035457250080344, 3.403200859788131, 4.003463721891949, 4.566682908108977, 5.427981269491059, 6.050192395724108, 6.606379151713716, 6.867988284617111, 7.889501076552205, 8.148468477808775, 8.835242488919892, 9.200692640599060, 9.711530581510352, 10.43909706200871, 10.79374361084678, 11.34441830753419, 11.96588218186651, 12.62992701412029, 13.04802280634032, 13.47194918693201, 13.85919477169545, 14.45321808331799
