| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 4·7-s + 8-s − 2·9-s + 3·11-s + 12-s − 6·13-s − 4·14-s + 16-s − 3·17-s − 2·18-s − 3·19-s − 4·21-s + 3·22-s − 2·23-s + 24-s − 6·26-s − 5·27-s − 4·28-s + 32-s + 3·33-s − 3·34-s − 2·36-s + 37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 1.51·7-s + 0.353·8-s − 2/3·9-s + 0.904·11-s + 0.288·12-s − 1.66·13-s − 1.06·14-s + 1/4·16-s − 0.727·17-s − 0.471·18-s − 0.688·19-s − 0.872·21-s + 0.639·22-s − 0.417·23-s + 0.204·24-s − 1.17·26-s − 0.962·27-s − 0.755·28-s + 0.176·32-s + 0.522·33-s − 0.514·34-s − 1/3·36-s + 0.164·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−9.026698028238458843593900329302, −8.017294356858624737125820472230, −7.02816789738378511430085190013, −6.48907771425825000167110429469, −5.70532021654462566329979908111, −4.58444402304127863143112825298, −3.72752891558989041392547388450, −2.88961457352644637819188364444, −2.17059239333815157645595147821, 0,
2.17059239333815157645595147821, 2.88961457352644637819188364444, 3.72752891558989041392547388450, 4.58444402304127863143112825298, 5.70532021654462566329979908111, 6.48907771425825000167110429469, 7.02816789738378511430085190013, 8.017294356858624737125820472230, 9.026698028238458843593900329302
