| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 7-s + 8-s + 9-s − 4·11-s + 12-s − 13-s − 14-s + 16-s − 17-s + 18-s − 4·19-s − 21-s − 4·22-s + 24-s − 26-s + 27-s − 28-s − 2·29-s + 32-s − 4·33-s − 34-s + 36-s − 2·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 1.20·11-s + 0.288·12-s − 0.277·13-s − 0.267·14-s + 1/4·16-s − 0.242·17-s + 0.235·18-s − 0.917·19-s − 0.218·21-s − 0.852·22-s + 0.204·24-s − 0.196·26-s + 0.192·27-s − 0.188·28-s − 0.371·29-s + 0.176·32-s − 0.696·33-s − 0.171·34-s + 1/6·36-s − 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 232050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 232050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−13.08087049321400, −12.76079197399897, −12.54165577873634, −11.80940989877034, −11.36655235771339, −10.76102434867635, −10.41325186640733, −10.02387341371441, −9.416741480085278, −8.911472860944112, −8.389606356794107, −7.929950299376526, −7.454628681756103, −6.954234049846065, −6.507725302858225, −5.906595221531417, −5.331624384276245, −5.016762500438576, −4.201876335389567, −3.980985501047153, −3.259200907880362, −2.697496463927768, −2.317316721594700, −1.800929613548908, −0.8140189603646805, 0,
0.8140189603646805, 1.800929613548908, 2.317316721594700, 2.697496463927768, 3.259200907880362, 3.980985501047153, 4.201876335389567, 5.016762500438576, 5.331624384276245, 5.906595221531417, 6.507725302858225, 6.954234049846065, 7.454628681756103, 7.929950299376526, 8.389606356794107, 8.911472860944112, 9.416741480085278, 10.02387341371441, 10.41325186640733, 10.76102434867635, 11.36655235771339, 11.80940989877034, 12.54165577873634, 12.76079197399897, 13.08087049321400
