| L(s) = 1 | + 3·7-s + 2·11-s − 13-s − 3·17-s − 2·19-s − 4·23-s − 2·29-s − 4·31-s − 5·37-s + 12·41-s + 7·43-s + 9·47-s + 2·49-s + 4·53-s + 6·59-s − 4·61-s − 10·67-s − 15·71-s + 2·73-s + 6·77-s + 8·79-s + 4·83-s − 2·89-s − 3·91-s − 10·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 1.13·7-s + 0.603·11-s − 0.277·13-s − 0.727·17-s − 0.458·19-s − 0.834·23-s − 0.371·29-s − 0.718·31-s − 0.821·37-s + 1.87·41-s + 1.06·43-s + 1.31·47-s + 2/7·49-s + 0.549·53-s + 0.781·59-s − 0.512·61-s − 1.22·67-s − 1.78·71-s + 0.234·73-s + 0.683·77-s + 0.900·79-s + 0.439·83-s − 0.211·89-s − 0.314·91-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−14.11954142028092, −13.75475613466470, −12.99992496382030, −12.64025600535737, −11.93451546916110, −11.73644417506448, −11.06875345239614, −10.65260583906647, −10.32866500925599, −9.387623556188683, −9.088035147173440, −8.689658143649667, −7.908108143143609, −7.648395335031903, −7.036810072471689, −6.472823484807156, −5.739795039646238, −5.481985660655470, −4.600298348169692, −4.211701805820835, −3.842381377796570, −2.828696731171151, −2.209732597712371, −1.716300561930430, −0.9625643739447346, 0,
0.9625643739447346, 1.716300561930430, 2.209732597712371, 2.828696731171151, 3.842381377796570, 4.211701805820835, 4.600298348169692, 5.481985660655470, 5.739795039646238, 6.472823484807156, 7.036810072471689, 7.648395335031903, 7.908108143143609, 8.689658143649667, 9.088035147173440, 9.387623556188683, 10.32866500925599, 10.65260583906647, 11.06875345239614, 11.73644417506448, 11.93451546916110, 12.64025600535737, 12.99992496382030, 13.75475613466470, 14.11954142028092
