| L(s) = 1 | − 2-s + 2·3-s + 4-s − 2·6-s − 8-s + 9-s + 2·12-s + 8·13-s + 16-s − 18-s − 2·24-s − 10·25-s − 8·26-s − 4·27-s − 32-s + 36-s + 4·37-s + 16·39-s + 24·47-s + 2·48-s + 10·50-s + 8·52-s + 4·54-s + 12·59-s − 16·61-s + 64-s − 72-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.816·6-s − 0.353·8-s + 1/3·9-s + 0.577·12-s + 2.21·13-s + 1/4·16-s − 0.235·18-s − 0.408·24-s − 2·25-s − 1.56·26-s − 0.769·27-s − 0.176·32-s + 1/6·36-s + 0.657·37-s + 2.56·39-s + 3.50·47-s + 0.288·48-s + 1.41·50-s + 1.10·52-s + 0.544·54-s + 1.56·59-s − 2.04·61-s + 1/8·64-s − 0.117·72-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 691488 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 691488 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.318542380\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.318542380\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( 1 + T \) |
| 3 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.573529742090186402738055178368, −7.88115201382147028507134927026, −7.55563465371540871894331307120, −7.39293674981761532961441986375, −6.43902889849944115155691598253, −6.12986038892916091786195943044, −5.83259812621372959122100845675, −5.25590580602395553760699468065, −4.24465289176609138955904372614, −3.87985229263682105648190826671, −3.56550924639142318669266713404, −2.84501304641458537647904209821, −2.25140513775369021989127014661, −1.67921161671410140538744564928, −0.818365395076437722978448101852,
0.818365395076437722978448101852, 1.67921161671410140538744564928, 2.25140513775369021989127014661, 2.84501304641458537647904209821, 3.56550924639142318669266713404, 3.87985229263682105648190826671, 4.24465289176609138955904372614, 5.25590580602395553760699468065, 5.83259812621372959122100845675, 6.12986038892916091786195943044, 6.43902889849944115155691598253, 7.39293674981761532961441986375, 7.55563465371540871894331307120, 7.88115201382147028507134927026, 8.573529742090186402738055178368
