| L(s) = 1 | − 2-s + 4-s + 5-s − 7-s − 8-s − 10-s − 11-s + 2·13-s + 14-s + 16-s − 2·19-s + 20-s + 22-s + 4·23-s + 25-s − 2·26-s − 28-s + 10·29-s − 4·31-s − 32-s − 35-s − 4·37-s + 2·38-s − 40-s − 6·41-s + 6·43-s − 44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.377·7-s − 0.353·8-s − 0.316·10-s − 0.301·11-s + 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.458·19-s + 0.223·20-s + 0.213·22-s + 0.834·23-s + 1/5·25-s − 0.392·26-s − 0.188·28-s + 1.85·29-s − 0.718·31-s − 0.176·32-s − 0.169·35-s − 0.657·37-s + 0.324·38-s − 0.158·40-s − 0.937·41-s + 0.914·43-s − 0.150·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.419032292\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.419032292\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−8.263783043118445658627637948939, −7.06277172352689287665702293193, −6.80497317566770523206681840269, −5.94177224906595788916007559231, −5.29957965689161254804884714096, −4.36707879462100514588224343088, −3.34711559818770167127130244291, −2.64536652094341162060297480995, −1.69961732288693581850333296609, −0.68508989103098903077156499148,
0.68508989103098903077156499148, 1.69961732288693581850333296609, 2.64536652094341162060297480995, 3.34711559818770167127130244291, 4.36707879462100514588224343088, 5.29957965689161254804884714096, 5.94177224906595788916007559231, 6.80497317566770523206681840269, 7.06277172352689287665702293193, 8.263783043118445658627637948939
