| L(s) = 1 | − 3-s + 9-s + 5·11-s + 2·13-s + 2·17-s − 6·19-s + 23-s − 27-s + 3·29-s − 4·31-s − 5·33-s − 5·37-s − 2·39-s − 4·41-s − 7·43-s + 10·47-s − 2·51-s + 2·53-s + 6·57-s − 10·59-s − 8·61-s + 7·67-s − 69-s − 3·71-s − 2·73-s − 11·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 1.50·11-s + 0.554·13-s + 0.485·17-s − 1.37·19-s + 0.208·23-s − 0.192·27-s + 0.557·29-s − 0.718·31-s − 0.870·33-s − 0.821·37-s − 0.320·39-s − 0.624·41-s − 1.06·43-s + 1.45·47-s − 0.280·51-s + 0.274·53-s + 0.794·57-s − 1.30·59-s − 1.02·61-s + 0.855·67-s − 0.120·69-s − 0.356·71-s − 0.234·73-s − 1.23·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−16.60246460375253, −15.76649804417081, −15.28918190384224, −14.73243819333832, −14.08808665012491, −13.68623843428486, −12.82086065969229, −12.42524945998085, −11.83366450813506, −11.34905068861642, −10.69678729481783, −10.24246959759356, −9.506910168949557, −8.787216376676440, −8.521416278241196, −7.525415357408664, −6.858452707553713, −6.394252495219791, −5.836487641382266, −5.078521282260279, −4.241196506995553, −3.834171874144027, −2.954054394569227, −1.778982927364904, −1.221813131386276, 0,
1.221813131386276, 1.778982927364904, 2.954054394569227, 3.834171874144027, 4.241196506995553, 5.078521282260279, 5.836487641382266, 6.394252495219791, 6.858452707553713, 7.525415357408664, 8.521416278241196, 8.787216376676440, 9.506910168949557, 10.24246959759356, 10.69678729481783, 11.34905068861642, 11.83366450813506, 12.42524945998085, 12.82086065969229, 13.68623843428486, 14.08808665012491, 14.73243819333832, 15.28918190384224, 15.76649804417081, 16.60246460375253
