| L(s) = 1 | + 2·3-s − 2·5-s + 5·7-s + 9-s + 6·11-s − 6·13-s − 4·15-s − 3·17-s + 8·19-s + 10·21-s + 2·23-s − 25-s − 4·27-s + 2·29-s + 9·31-s + 12·33-s − 10·35-s − 4·37-s − 12·39-s − 6·41-s + 12·43-s − 2·45-s − 3·47-s + 18·49-s − 6·51-s + 3·53-s − 12·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s + 1.88·7-s + 1/3·9-s + 1.80·11-s − 1.66·13-s − 1.03·15-s − 0.727·17-s + 1.83·19-s + 2.18·21-s + 0.417·23-s − 1/5·25-s − 0.769·27-s + 0.371·29-s + 1.61·31-s + 2.08·33-s − 1.69·35-s − 0.657·37-s − 1.92·39-s − 0.937·41-s + 1.82·43-s − 0.298·45-s − 0.437·47-s + 18/7·49-s − 0.840·51-s + 0.412·53-s − 1.61·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.227247175\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.227247175\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 6571 | \( 1 + T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 5 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 15 T + p T^{2} \) |
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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
−13.96036663659987, −13.66196508964497, −12.45719082999505, −12.14384315245348, −11.77255325163238, −11.32823223013953, −11.08819414187222, −10.02700545529568, −9.722430440071853, −9.048663601893880, −8.753248764007925, −8.198039048917873, −7.766379684537998, −7.350100568422867, −7.004184239241558, −6.191687651249469, −5.256543698943850, −4.901263293152584, −4.308527691662332, −3.937257283717921, −3.230684805574466, −2.602735710289077, −2.011146449372398, −1.351140720885502, −0.6965568344241447,
0.6965568344241447, 1.351140720885502, 2.011146449372398, 2.602735710289077, 3.230684805574466, 3.937257283717921, 4.308527691662332, 4.901263293152584, 5.256543698943850, 6.191687651249469, 7.004184239241558, 7.350100568422867, 7.766379684537998, 8.198039048917873, 8.753248764007925, 9.048663601893880, 9.722430440071853, 10.02700545529568, 11.08819414187222, 11.32823223013953, 11.77255325163238, 12.14384315245348, 12.45719082999505, 13.66196508964497, 13.96036663659987
