| L(s) = 1 | − 3-s + 5-s − 2·7-s + 9-s + 2·13-s − 15-s − 6·17-s + 2·21-s − 6·23-s + 25-s − 27-s − 4·29-s − 2·35-s + 10·37-s − 2·39-s + 8·41-s + 2·43-s + 45-s + 2·47-s − 3·49-s + 6·51-s − 2·53-s − 14·61-s − 2·63-s + 2·65-s − 4·67-s + 6·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s + 0.554·13-s − 0.258·15-s − 1.45·17-s + 0.436·21-s − 1.25·23-s + 1/5·25-s − 0.192·27-s − 0.742·29-s − 0.338·35-s + 1.64·37-s − 0.320·39-s + 1.24·41-s + 0.304·43-s + 0.149·45-s + 0.291·47-s − 3/7·49-s + 0.840·51-s − 0.274·53-s − 1.79·61-s − 0.251·63-s + 0.248·65-s − 0.488·67-s + 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 346560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 346560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−12.77069841227180, −12.40633758736576, −11.88838810639051, −11.32114468828303, −10.91240844140777, −10.71045248501089, −10.01834279196882, −9.542757940661712, −9.293744749150279, −8.837524880120246, −8.097020637130353, −7.762530935466804, −7.151255258484986, −6.562486980552241, −6.217927166117728, −5.954620059130478, −5.433882714130920, −4.679332175193812, −4.284166766014956, −3.852625786573656, −3.173947100329994, −2.511140818045928, −2.076015340508380, −1.406279438522073, −0.6396882605074185, 0,
0.6396882605074185, 1.406279438522073, 2.076015340508380, 2.511140818045928, 3.173947100329994, 3.852625786573656, 4.284166766014956, 4.679332175193812, 5.433882714130920, 5.954620059130478, 6.217927166117728, 6.562486980552241, 7.151255258484986, 7.762530935466804, 8.097020637130353, 8.837524880120246, 9.293744749150279, 9.542757940661712, 10.01834279196882, 10.71045248501089, 10.91240844140777, 11.32114468828303, 11.88838810639051, 12.40633758736576, 12.77069841227180
