| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 7-s − 8-s − 2·9-s + 10-s + 12-s + 5·13-s + 14-s − 15-s + 16-s + 3·17-s + 2·18-s − 19-s − 20-s − 21-s + 6·23-s − 24-s + 25-s − 5·26-s − 5·27-s − 28-s + 29-s + 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s − 2/3·9-s + 0.316·10-s + 0.288·12-s + 1.38·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.727·17-s + 0.471·18-s − 0.229·19-s − 0.223·20-s − 0.218·21-s + 1.25·23-s − 0.204·24-s + 1/5·25-s − 0.980·26-s − 0.962·27-s − 0.188·28-s + 0.185·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| 139 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−15.08245724237234, −14.52540842514501, −14.19043269449803, −13.37902202048384, −13.01527033659800, −12.49240145489634, −11.61707891107850, −11.39854978983076, −10.86069684407156, −10.25861859658949, −9.678325819940330, −9.009348165033821, −8.676551353971095, −8.289879579898745, −7.622364986279671, −7.126710334684297, −6.458055610372899, −5.848663080147633, −5.358063355799316, −4.373221757511192, −3.639885980585048, −3.164961442584680, −2.674270552264366, −1.663532071015202, −0.9773325371848149, 0,
0.9773325371848149, 1.663532071015202, 2.674270552264366, 3.164961442584680, 3.639885980585048, 4.373221757511192, 5.358063355799316, 5.848663080147633, 6.458055610372899, 7.126710334684297, 7.622364986279671, 8.289879579898745, 8.676551353971095, 9.009348165033821, 9.678325819940330, 10.25861859658949, 10.86069684407156, 11.39854978983076, 11.61707891107850, 12.49240145489634, 13.01527033659800, 13.37902202048384, 14.19043269449803, 14.52540842514501, 15.08245724237234
