| L(s) = 1 | + 3·3-s − 16·7-s + 9·9-s + 24·11-s + 14·13-s + 18·17-s + 36·19-s − 48·21-s − 104·23-s + 27·27-s − 250·29-s − 28·31-s + 72·33-s + 54·37-s + 42·39-s + 354·41-s − 228·43-s − 408·47-s − 87·49-s + 54·51-s − 262·53-s + 108·57-s − 64·59-s + 374·61-s − 144·63-s − 300·67-s − 312·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.863·7-s + 1/3·9-s + 0.657·11-s + 0.298·13-s + 0.256·17-s + 0.434·19-s − 0.498·21-s − 0.942·23-s + 0.192·27-s − 1.60·29-s − 0.162·31-s + 0.379·33-s + 0.239·37-s + 0.172·39-s + 1.34·41-s − 0.808·43-s − 1.26·47-s − 0.253·49-s + 0.148·51-s − 0.679·53-s + 0.250·57-s − 0.141·59-s + 0.785·61-s − 0.287·63-s − 0.547·67-s − 0.544·69-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
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 16 T + p^{3} T^{2} \) |
| 11 | \( 1 - 24 T + p^{3} T^{2} \) |
| 13 | \( 1 - 14 T + p^{3} T^{2} \) |
| 17 | \( 1 - 18 T + p^{3} T^{2} \) |
| 19 | \( 1 - 36 T + p^{3} T^{2} \) |
| 23 | \( 1 + 104 T + p^{3} T^{2} \) |
| 29 | \( 1 + 250 T + p^{3} T^{2} \) |
| 31 | \( 1 + 28 T + p^{3} T^{2} \) |
| 37 | \( 1 - 54 T + p^{3} T^{2} \) |
| 41 | \( 1 - 354 T + p^{3} T^{2} \) |
| 43 | \( 1 + 228 T + p^{3} T^{2} \) |
| 47 | \( 1 + 408 T + p^{3} T^{2} \) |
| 53 | \( 1 + 262 T + p^{3} T^{2} \) |
| 59 | \( 1 + 64 T + p^{3} T^{2} \) |
| 61 | \( 1 - 374 T + p^{3} T^{2} \) |
| 67 | \( 1 + 300 T + p^{3} T^{2} \) |
| 71 | \( 1 - 1016 T + p^{3} T^{2} \) |
| 73 | \( 1 + 274 T + p^{3} T^{2} \) |
| 79 | \( 1 - 788 T + p^{3} T^{2} \) |
| 83 | \( 1 - 396 T + p^{3} T^{2} \) |
| 89 | \( 1 - 786 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1086 T + p^{3} T^{2} \) |
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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
−8.159668433281619779180812148575, −7.58353796403289798981957866432, −6.62953575573381251002114662900, −6.06418783944700619007387406110, −5.04921019975582846922510353985, −3.85197864762603733614338465303, −3.46827251064966672144131187908, −2.34738798446449786263077167333, −1.31886441216474891540001272383, 0,
1.31886441216474891540001272383, 2.34738798446449786263077167333, 3.46827251064966672144131187908, 3.85197864762603733614338465303, 5.04921019975582846922510353985, 6.06418783944700619007387406110, 6.62953575573381251002114662900, 7.58353796403289798981957866432, 8.159668433281619779180812148575
