| 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
| \(L(1)\) |
\(\approx\) |
\(0.6529682773\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6529682773\) |
| \(L(\frac{3}{2})\) |
|
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
−12.62097584969239, −11.96429518599371, −11.71341512384430, −11.11633351222142, −10.80515352768399, −10.40753152103509, −9.920116335821422, −9.336959848220944, −8.924068052367779, −8.479825612726216, −8.031631398280515, −7.347537210086681, −6.877285361610229, −6.609752387784246, −6.108224505761096, −5.279009929192680, −5.060501016236191, −4.750401526103312, −4.121979758592642, −3.480385990372552, −2.756651570537176, −2.331172963012092, −1.507291287855144, −1.347037936427132, −0.2114494414027702,
0.2114494414027702, 1.347037936427132, 1.507291287855144, 2.331172963012092, 2.756651570537176, 3.480385990372552, 4.121979758592642, 4.750401526103312, 5.060501016236191, 5.279009929192680, 6.108224505761096, 6.609752387784246, 6.877285361610229, 7.347537210086681, 8.031631398280515, 8.479825612726216, 8.924068052367779, 9.336959848220944, 9.920116335821422, 10.40753152103509, 10.80515352768399, 11.11633351222142, 11.71341512384430, 11.96429518599371, 12.62097584969239
