| L(s) = 1 | − 2-s + 4-s − 4·7-s − 8-s + 2·13-s + 4·14-s + 16-s + 2·17-s + 4·19-s − 2·26-s − 4·28-s − 2·29-s − 32-s − 2·34-s − 10·37-s − 4·38-s + 6·41-s + 43-s − 8·47-s + 9·49-s + 2·52-s + 2·53-s + 4·56-s + 2·58-s + 2·61-s + 64-s + 4·67-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.51·7-s − 0.353·8-s + 0.554·13-s + 1.06·14-s + 1/4·16-s + 0.485·17-s + 0.917·19-s − 0.392·26-s − 0.755·28-s − 0.371·29-s − 0.176·32-s − 0.342·34-s − 1.64·37-s − 0.648·38-s + 0.937·41-s + 0.152·43-s − 1.16·47-s + 9/7·49-s + 0.277·52-s + 0.274·53-s + 0.534·56-s + 0.262·58-s + 0.256·61-s + 1/8·64-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 43 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 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 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 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
−16.00278279128075, −15.76340600203178, −15.02785289800028, −14.36423285451882, −13.73666712868001, −13.18926376208839, −12.66071161996854, −12.11716138034180, −11.53234938272481, −10.91265657987798, −10.14711149016685, −9.951147556958538, −9.223132569585334, −8.854718079641130, −8.109693562093683, −7.374046679499283, −6.966040129966168, −6.240901535174999, −5.793237181973799, −5.060053954866871, −3.960973103908890, −3.334410200488630, −2.895386457117688, −1.852496678211092, −0.9400668082633428, 0,
0.9400668082633428, 1.852496678211092, 2.895386457117688, 3.334410200488630, 3.960973103908890, 5.060053954866871, 5.793237181973799, 6.240901535174999, 6.966040129966168, 7.374046679499283, 8.109693562093683, 8.854718079641130, 9.223132569585334, 9.951147556958538, 10.14711149016685, 10.91265657987798, 11.53234938272481, 12.11716138034180, 12.66071161996854, 13.18926376208839, 13.73666712868001, 14.36423285451882, 15.02785289800028, 15.76340600203178, 16.00278279128075
