| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 7-s − 8-s + 9-s − 10-s − 2·11-s − 12-s − 13-s + 14-s − 15-s + 16-s − 18-s + 4·19-s + 20-s + 21-s + 2·22-s + 24-s + 25-s + 26-s − 27-s − 28-s − 2·29-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 + 1/3·9-s − 0.316·10-s − 0.603·11-s − 0.288·12-s − 0.277·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s − 0.235·18-s + 0.917·19-s + 0.223·20-s + 0.218·21-s + 0.426·22-s + 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s − 0.188·28-s − 0.371·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 372810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 372810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 17 | \( 1 \) |
| 43 | \( 1 - T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 11 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−12.59017997071622, −12.06821474911614, −11.94284762324259, −11.24894827777336, −10.84962151096347, −10.34067758329918, −10.09315191821395, −9.528193448936955, −9.302706487687303, −8.653245424538422, −8.109917822468544, −7.716665012410668, −7.224123820208453, −6.666175272341436, −6.391265860760873, −5.748706883845756, −5.343405891998134, −4.934083520476156, −4.264956710727610, −3.639260239818820, −2.946313301417976, −2.577663172853052, −1.955272569398862, −1.212756085837257, −0.7339316365140275, 0,
0.7339316365140275, 1.212756085837257, 1.955272569398862, 2.577663172853052, 2.946313301417976, 3.639260239818820, 4.264956710727610, 4.934083520476156, 5.343405891998134, 5.748706883845756, 6.391265860760873, 6.666175272341436, 7.224123820208453, 7.716665012410668, 8.109917822468544, 8.653245424538422, 9.302706487687303, 9.528193448936955, 10.09315191821395, 10.34067758329918, 10.84962151096347, 11.24894827777336, 11.94284762324259, 12.06821474911614, 12.59017997071622
