| L(s) = 1 | − 2-s + 4-s − 8-s − 5·9-s + 4·11-s + 2·13-s + 16-s + 6·17-s + 5·18-s − 19-s − 4·22-s + 6·25-s − 2·26-s + 10·29-s + 16·31-s − 32-s − 6·34-s − 5·36-s + 4·37-s + 38-s + 8·43-s + 4·44-s − 5·49-s − 6·50-s + 2·52-s + 2·53-s − 10·58-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 5/3·9-s + 1.20·11-s + 0.554·13-s + 1/4·16-s + 1.45·17-s + 1.17·18-s − 0.229·19-s − 0.852·22-s + 6/5·25-s − 0.392·26-s + 1.85·29-s + 2.87·31-s − 0.176·32-s − 1.02·34-s − 5/6·36-s + 0.657·37-s + 0.162·38-s + 1.21·43-s + 0.603·44-s − 5/7·49-s − 0.848·50-s + 0.277·52-s + 0.274·53-s − 1.31·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 877952 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 877952 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.800839115\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.800839115\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 19 | $C_1$ | \( 1 + T \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.336890921646432967363618675577, −7.957421913838810837694491246300, −7.46242727862265190447093907017, −6.67486851525401293685577568233, −6.41602681477419049736409749590, −6.16947011382354679761132974999, −5.70311644291868843897751049539, −4.89838836690724431377597203004, −4.71396131109660693238138315060, −3.78873474603954853324187046999, −3.35219753314436267860210658260, −2.72199195247203046939301106692, −2.45747388367035991238545338827, −1.03106032747078173831338248002, −0.980313161146313482644993247523,
0.980313161146313482644993247523, 1.03106032747078173831338248002, 2.45747388367035991238545338827, 2.72199195247203046939301106692, 3.35219753314436267860210658260, 3.78873474603954853324187046999, 4.71396131109660693238138315060, 4.89838836690724431377597203004, 5.70311644291868843897751049539, 6.16947011382354679761132974999, 6.41602681477419049736409749590, 6.67486851525401293685577568233, 7.46242727862265190447093907017, 7.957421913838810837694491246300, 8.336890921646432967363618675577
