| L(s) = 1 | + 2.90·3-s + 1.52·7-s + 5.42·9-s + 0.903·11-s − 2.42·13-s − 2.28·17-s − 0.474·19-s + 4.42·21-s + 2.90·23-s + 7.05·27-s + 29-s + 5.33·31-s + 2.62·33-s + 3.52·37-s − 7.05·39-s + 4.62·41-s + 12.7·43-s + 1.65·47-s − 4.67·49-s − 6.62·51-s + 2.13·53-s − 1.37·57-s − 7.18·59-s − 7.67·61-s + 8.28·63-s + 2.04·67-s + 8.42·69-s + ⋯ |
| L(s) = 1 | + 1.67·3-s + 0.576·7-s + 1.80·9-s + 0.272·11-s − 0.673·13-s − 0.553·17-s − 0.108·19-s + 0.966·21-s + 0.605·23-s + 1.35·27-s + 0.185·29-s + 0.957·31-s + 0.456·33-s + 0.579·37-s − 1.12·39-s + 0.721·41-s + 1.93·43-s + 0.241·47-s − 0.667·49-s − 0.927·51-s + 0.293·53-s − 0.182·57-s − 0.935·59-s − 0.982·61-s + 1.04·63-s + 0.249·67-s + 1.01·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.260640096\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.260640096\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 - 2.90T + 3T^{2} \) |
| 7 | \( 1 - 1.52T + 7T^{2} \) |
| 11 | \( 1 - 0.903T + 11T^{2} \) |
| 13 | \( 1 + 2.42T + 13T^{2} \) |
| 17 | \( 1 + 2.28T + 17T^{2} \) |
| 19 | \( 1 + 0.474T + 19T^{2} \) |
| 23 | \( 1 - 2.90T + 23T^{2} \) |
| 31 | \( 1 - 5.33T + 31T^{2} \) |
| 37 | \( 1 - 3.52T + 37T^{2} \) |
| 41 | \( 1 - 4.62T + 41T^{2} \) |
| 43 | \( 1 - 12.7T + 43T^{2} \) |
| 47 | \( 1 - 1.65T + 47T^{2} \) |
| 53 | \( 1 - 2.13T + 53T^{2} \) |
| 59 | \( 1 + 7.18T + 59T^{2} \) |
| 61 | \( 1 + 7.67T + 61T^{2} \) |
| 67 | \( 1 - 2.04T + 67T^{2} \) |
| 71 | \( 1 - 3.18T + 71T^{2} \) |
| 73 | \( 1 - 8.90T + 73T^{2} \) |
| 79 | \( 1 - 5.19T + 79T^{2} \) |
| 83 | \( 1 - 15.6T + 83T^{2} \) |
| 89 | \( 1 - 11.8T + 89T^{2} \) |
| 97 | \( 1 + 3.39T + 97T^{2} \) |
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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
−7.912418471888478987258128782241, −7.79036263769730953938981108028, −6.91400647340674355972332874351, −6.12067714675722880024549564354, −4.90158723438524163152522226910, −4.38241663293421317983369878591, −3.56470687722616393654532493944, −2.65091895868458916114258641261, −2.17806929521485719404300817977, −1.06017737021434199453814467594,
1.06017737021434199453814467594, 2.17806929521485719404300817977, 2.65091895868458916114258641261, 3.56470687722616393654532493944, 4.38241663293421317983369878591, 4.90158723438524163152522226910, 6.12067714675722880024549564354, 6.91400647340674355972332874351, 7.79036263769730953938981108028, 7.912418471888478987258128782241
