| L(s) = 1 | + 3·2-s + 3-s + 4-s − 6·5-s + 3·6-s + 7·7-s − 21·8-s − 26·9-s − 18·10-s − 30·11-s + 12-s − 22·13-s + 21·14-s − 6·15-s − 71·16-s − 78·18-s + 83·19-s − 6·20-s + 7·21-s − 90·22-s − 48·23-s − 21·24-s − 89·25-s − 66·26-s − 53·27-s + 7·28-s + 15·29-s + ⋯ |
| L(s) = 1 | + 1.06·2-s + 0.192·3-s + 1/8·4-s − 0.536·5-s + 0.204·6-s + 0.377·7-s − 0.928·8-s − 0.962·9-s − 0.569·10-s − 0.822·11-s + 0.0240·12-s − 0.469·13-s + 0.400·14-s − 0.103·15-s − 1.10·16-s − 1.02·18-s + 1.00·19-s − 0.0670·20-s + 0.0727·21-s − 0.872·22-s − 0.435·23-s − 0.178·24-s − 0.711·25-s − 0.497·26-s − 0.377·27-s + 0.0472·28-s + 0.0960·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2023 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.910699772\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.910699772\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - p T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 3 T + p^{3} T^{2} \) |
| 3 | \( 1 - T + p^{3} T^{2} \) |
| 5 | \( 1 + 6 T + p^{3} T^{2} \) |
| 11 | \( 1 + 30 T + p^{3} T^{2} \) |
| 13 | \( 1 + 22 T + p^{3} T^{2} \) |
| 19 | \( 1 - 83 T + p^{3} T^{2} \) |
| 23 | \( 1 + 48 T + p^{3} T^{2} \) |
| 29 | \( 1 - 15 T + p^{3} T^{2} \) |
| 31 | \( 1 + 7 T + p^{3} T^{2} \) |
| 37 | \( 1 - 50 T + p^{3} T^{2} \) |
| 41 | \( 1 - 360 T + p^{3} T^{2} \) |
| 43 | \( 1 - 68 T + p^{3} T^{2} \) |
| 47 | \( 1 + 27 T + p^{3} T^{2} \) |
| 53 | \( 1 - 213 T + p^{3} T^{2} \) |
| 59 | \( 1 - 189 T + p^{3} T^{2} \) |
| 61 | \( 1 - 314 T + p^{3} T^{2} \) |
| 67 | \( 1 - 314 T + p^{3} T^{2} \) |
| 71 | \( 1 + 804 T + p^{3} T^{2} \) |
| 73 | \( 1 + 448 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1060 T + p^{3} T^{2} \) |
| 83 | \( 1 - 873 T + p^{3} T^{2} \) |
| 89 | \( 1 - 270 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1130 T + p^{3} T^{2} \) |
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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
−8.700747390928924328226715839680, −7.926653811034240853794489827594, −7.32594406418970882202414097246, −6.03255031374341326859937629720, −5.49880153509838802864598966109, −4.74424126418896486657960037920, −3.90156937170694219925473101549, −3.05111356095680159879296359130, −2.30618882105488553885934667513, −0.50681231527312268211104195951,
0.50681231527312268211104195951, 2.30618882105488553885934667513, 3.05111356095680159879296359130, 3.90156937170694219925473101549, 4.74424126418896486657960037920, 5.49880153509838802864598966109, 6.03255031374341326859937629720, 7.32594406418970882202414097246, 7.926653811034240853794489827594, 8.700747390928924328226715839680
