| L(s) = 1 | + 2·2-s − 4·3-s + 4·4-s − 8·6-s + 4·7-s + 8·8-s − 11·9-s − 12·11-s − 16·12-s + 58·13-s + 8·14-s + 16·16-s − 17·17-s − 22·18-s − 52·19-s − 16·21-s − 24·22-s − 84·23-s − 32·24-s + 116·26-s + 152·27-s + 16·28-s − 246·29-s + 68·31-s + 32·32-s + 48·33-s − 34·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.769·3-s + 1/2·4-s − 0.544·6-s + 0.215·7-s + 0.353·8-s − 0.407·9-s − 0.328·11-s − 0.384·12-s + 1.23·13-s + 0.152·14-s + 1/4·16-s − 0.242·17-s − 0.288·18-s − 0.627·19-s − 0.166·21-s − 0.232·22-s − 0.761·23-s − 0.272·24-s + 0.874·26-s + 1.08·27-s + 0.107·28-s − 1.57·29-s + 0.393·31-s + 0.176·32-s + 0.253·33-s − 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p T \) |
| 5 | \( 1 \) |
| 17 | \( 1 + p T \) |
| good | 3 | \( 1 + 4 T + p^{3} T^{2} \) |
| 7 | \( 1 - 4 T + p^{3} T^{2} \) |
| 11 | \( 1 + 12 T + p^{3} T^{2} \) |
| 13 | \( 1 - 58 T + p^{3} T^{2} \) |
| 19 | \( 1 + 52 T + p^{3} T^{2} \) |
| 23 | \( 1 + 84 T + p^{3} T^{2} \) |
| 29 | \( 1 + 246 T + p^{3} T^{2} \) |
| 31 | \( 1 - 68 T + p^{3} T^{2} \) |
| 37 | \( 1 - 358 T + p^{3} T^{2} \) |
| 41 | \( 1 + 78 T + p^{3} T^{2} \) |
| 43 | \( 1 - 412 T + p^{3} T^{2} \) |
| 47 | \( 1 + 408 T + p^{3} T^{2} \) |
| 53 | \( 1 + 750 T + p^{3} T^{2} \) |
| 59 | \( 1 + 420 T + p^{3} T^{2} \) |
| 61 | \( 1 + 190 T + p^{3} T^{2} \) |
| 67 | \( 1 + 596 T + p^{3} T^{2} \) |
| 71 | \( 1 - 324 T + p^{3} T^{2} \) |
| 73 | \( 1 + 1010 T + p^{3} T^{2} \) |
| 79 | \( 1 - 164 T + p^{3} T^{2} \) |
| 83 | \( 1 + 588 T + p^{3} T^{2} \) |
| 89 | \( 1 + 486 T + p^{3} T^{2} \) |
| 97 | \( 1 - 718 T + p^{3} T^{2} \) |
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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
−9.449420793849084899338315899574, −8.385976876651784947256139852562, −7.60367518022969037860989535103, −6.18620806235024070495678264655, −6.08286571968011613966188160740, −4.93989611663481390263539840334, −4.05946912593849336500733119523, −2.90494046951231390001272532003, −1.55946700605040616147598116556, 0,
1.55946700605040616147598116556, 2.90494046951231390001272532003, 4.05946912593849336500733119523, 4.93989611663481390263539840334, 6.08286571968011613966188160740, 6.18620806235024070495678264655, 7.60367518022969037860989535103, 8.385976876651784947256139852562, 9.449420793849084899338315899574
