| L(s) = 1 | − 2-s + 4·3-s + 4-s − 4·6-s − 8-s + 6·9-s + 4·12-s + 16-s − 12·17-s − 6·18-s − 4·19-s − 4·24-s − 10·25-s − 4·27-s − 32-s + 12·34-s + 6·36-s + 4·38-s − 12·41-s + 16·43-s + 4·48-s + 10·50-s − 48·51-s + 4·54-s − 16·57-s + 12·59-s + 64-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 2.30·3-s + 1/2·4-s − 1.63·6-s − 0.353·8-s + 2·9-s + 1.15·12-s + 1/4·16-s − 2.91·17-s − 1.41·18-s − 0.917·19-s − 0.816·24-s − 2·25-s − 0.769·27-s − 0.176·32-s + 2.05·34-s + 36-s + 0.648·38-s − 1.87·41-s + 2.43·43-s + 0.577·48-s + 1.41·50-s − 6.72·51-s + 0.544·54-s − 2.11·57-s + 1.56·59-s + 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 307328 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 307328 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( 1 + T \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 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
−8.624667079434487291694904422005, −8.408228795468761224431793416064, −7.68957389720506083723150978195, −7.55563465371540871894331307120, −6.96175265373852036821224883683, −6.17368924384109906924363362759, −6.12986038892916091786195943044, −5.06701824847582369879735343431, −4.31547623044891473234039816601, −3.87985229263682105648190826671, −3.42443184461883656887541100921, −2.49739457868022289115900838847, −2.25140513775369021989127014661, −1.90996831082337002056034743405, 0,
1.90996831082337002056034743405, 2.25140513775369021989127014661, 2.49739457868022289115900838847, 3.42443184461883656887541100921, 3.87985229263682105648190826671, 4.31547623044891473234039816601, 5.06701824847582369879735343431, 6.12986038892916091786195943044, 6.17368924384109906924363362759, 6.96175265373852036821224883683, 7.55563465371540871894331307120, 7.68957389720506083723150978195, 8.408228795468761224431793416064, 8.624667079434487291694904422005
