| L(s) = 1 | + 2·2-s − 2·3-s + 2·4-s − 4·6-s + 9-s − 4·12-s − 6·13-s − 4·16-s + 2·18-s − 6·25-s − 12·26-s + 4·27-s + 8·29-s − 12·31-s − 8·32-s + 2·36-s + 12·39-s + 8·48-s − 10·49-s − 12·50-s − 12·52-s + 8·54-s + 16·58-s − 24·62-s − 8·64-s − 8·67-s − 16·71-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 4-s − 1.63·6-s + 1/3·9-s − 1.15·12-s − 1.66·13-s − 16-s + 0.471·18-s − 6/5·25-s − 2.35·26-s + 0.769·27-s + 1.48·29-s − 2.15·31-s − 1.41·32-s + 1/3·36-s + 1.92·39-s + 1.15·48-s − 1.42·49-s − 1.69·50-s − 1.66·52-s + 1.08·54-s + 2.10·58-s − 3.04·62-s − 64-s − 0.977·67-s − 1.89·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 66564 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66564 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 3 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 43 | $C_2$ | \( 1 + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 39 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 125 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
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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
−9.684924122404473196473736310410, −9.307643029985204939438377253864, −8.620397011206227065181238031367, −7.899266047357790077870550388936, −7.23290245548422185741418791214, −6.93187925707354214747909238414, −6.13403334444091774717347862798, −5.89009893484286346541297709798, −5.23822359913803714588681626810, −4.81254717205194114342212792391, −4.40811315459086708871774292497, −3.52989136512868299764841720377, −2.83522495149789278874458619158, −1.95913848790194039121416340901, 0,
1.95913848790194039121416340901, 2.83522495149789278874458619158, 3.52989136512868299764841720377, 4.40811315459086708871774292497, 4.81254717205194114342212792391, 5.23822359913803714588681626810, 5.89009893484286346541297709798, 6.13403334444091774717347862798, 6.93187925707354214747909238414, 7.23290245548422185741418791214, 7.899266047357790077870550388936, 8.620397011206227065181238031367, 9.307643029985204939438377253864, 9.684924122404473196473736310410
