| L(s) = 1 | − 3-s + 5-s + 4·7-s + 9-s + 2·13-s − 15-s + 2·17-s − 4·19-s − 4·21-s + 25-s − 27-s − 2·29-s + 4·35-s − 10·37-s − 2·39-s − 6·41-s + 43-s + 45-s − 8·47-s + 9·49-s − 2·51-s − 2·53-s + 4·57-s − 2·61-s + 4·63-s + 2·65-s + 4·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.51·7-s + 1/3·9-s + 0.554·13-s − 0.258·15-s + 0.485·17-s − 0.917·19-s − 0.872·21-s + 1/5·25-s − 0.192·27-s − 0.371·29-s + 0.676·35-s − 1.64·37-s − 0.320·39-s − 0.937·41-s + 0.152·43-s + 0.149·45-s − 1.16·47-s + 9/7·49-s − 0.280·51-s − 0.274·53-s + 0.529·57-s − 0.256·61-s + 0.503·63-s + 0.248·65-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 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
−15.00560455281944, −14.44460298040716, −13.99510330041890, −13.56928094552481, −12.78468115390152, −12.48354866574430, −11.73431004649986, −11.32114422242853, −10.94463891452736, −10.31026186114969, −9.974686338928951, −9.094267386022543, −8.571707341802885, −8.148068149065218, −7.540117319706188, −6.840136796634084, −6.343130915155094, −5.646287717916333, −5.130395483648752, −4.750058376078101, −3.970142901279768, −3.348488033180142, −2.271177803245036, −1.690315716129054, −1.175177382089044, 0,
1.175177382089044, 1.690315716129054, 2.271177803245036, 3.348488033180142, 3.970142901279768, 4.750058376078101, 5.130395483648752, 5.646287717916333, 6.343130915155094, 6.840136796634084, 7.540117319706188, 8.148068149065218, 8.571707341802885, 9.094267386022543, 9.974686338928951, 10.31026186114969, 10.94463891452736, 11.32114422242853, 11.73431004649986, 12.48354866574430, 12.78468115390152, 13.56928094552481, 13.99510330041890, 14.44460298040716, 15.00560455281944
