| L(s) = 1 | + 3·3-s + 22·7-s + 9·9-s − 18.8·11-s − 18.8·13-s − 132.·17-s − 113.·19-s + 66·21-s + 160·23-s + 27·27-s + 128·29-s + 75.4·31-s − 56.6·33-s + 18.8·37-s − 56.6·39-s − 358·41-s + 172·43-s − 4·47-s + 141·49-s − 396.·51-s + 660.·53-s − 339.·57-s + 735.·59-s − 14·61-s + 198·63-s + 848·67-s + 480·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.18·7-s + 0.333·9-s − 0.517·11-s − 0.402·13-s − 1.88·17-s − 1.36·19-s + 0.685·21-s + 1.45·23-s + 0.192·27-s + 0.819·29-s + 0.437·31-s − 0.298·33-s + 0.0838·37-s − 0.232·39-s − 1.36·41-s + 0.609·43-s − 0.0124·47-s + 0.411·49-s − 1.08·51-s + 1.71·53-s − 0.789·57-s + 1.62·59-s − 0.0293·61-s + 0.395·63-s + 1.54·67-s + 0.837·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.891865585\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.891865585\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 22T + 343T^{2} \) |
| 11 | \( 1 + 18.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 18.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 132.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 113.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 160T + 1.21e4T^{2} \) |
| 29 | \( 1 - 128T + 2.43e4T^{2} \) |
| 31 | \( 1 - 75.4T + 2.97e4T^{2} \) |
| 37 | \( 1 - 18.8T + 5.06e4T^{2} \) |
| 41 | \( 1 + 358T + 6.89e4T^{2} \) |
| 43 | \( 1 - 172T + 7.95e4T^{2} \) |
| 47 | \( 1 + 4T + 1.03e5T^{2} \) |
| 53 | \( 1 - 660.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 735.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 14T + 2.26e5T^{2} \) |
| 67 | \( 1 - 848T + 3.00e5T^{2} \) |
| 71 | \( 1 - 641.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.16e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 75.4T + 4.93e5T^{2} \) |
| 83 | \( 1 - 596T + 5.71e5T^{2} \) |
| 89 | \( 1 - 750T + 7.04e5T^{2} \) |
| 97 | \( 1 + 528.T + 9.12e5T^{2} \) |
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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
−8.500241969813315964248383812981, −8.117544103397376689614046341764, −7.02668426353619637851311334043, −6.56381817281832899953178840184, −5.15228107374974199916009248876, −4.70915459903544861228441403993, −3.85227827621379414317353091255, −2.48939184404913320820985350470, −2.08382785758785839528011326315, −0.72011646640589936758896360802,
0.72011646640589936758896360802, 2.08382785758785839528011326315, 2.48939184404913320820985350470, 3.85227827621379414317353091255, 4.70915459903544861228441403993, 5.15228107374974199916009248876, 6.56381817281832899953178840184, 7.02668426353619637851311334043, 8.117544103397376689614046341764, 8.500241969813315964248383812981
