| L(s) = 1 | − 2·3-s − 3·9-s − 10·11-s + 6·17-s − 12·19-s + 25-s + 14·27-s + 20·33-s + 16·41-s + 12·43-s + 49-s − 12·51-s + 24·57-s + 16·59-s − 8·67-s + 20·73-s − 2·75-s − 4·81-s − 24·83-s − 32·89-s + 14·97-s + 30·99-s + 4·107-s − 12·113-s + 53·121-s − 32·123-s + 127-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 9-s − 3.01·11-s + 1.45·17-s − 2.75·19-s + 1/5·25-s + 2.69·27-s + 3.48·33-s + 2.49·41-s + 1.82·43-s + 1/7·49-s − 1.68·51-s + 3.17·57-s + 2.08·59-s − 0.977·67-s + 2.34·73-s − 0.230·75-s − 4/9·81-s − 2.63·83-s − 3.39·89-s + 1.42·97-s + 3.01·99-s + 0.386·107-s − 1.12·113-s + 4.81·121-s − 2.88·123-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 627200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 627200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
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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
−8.321037332106296729315848288758, −7.61068895852562797070369378134, −7.47320461628172485062800558626, −6.62516017863013169349321112564, −6.17106315121314565062659046983, −5.71012569325521424791956643699, −5.45734437575009036836059962399, −5.22465963352202663879592762973, −4.44994957570517170152202331319, −4.06182673812785905054132006247, −2.98004661899199231115437847100, −2.60467600555756229149551481858, −2.31436414126049041586401831975, −0.73530382675733263946372082289, 0,
0.73530382675733263946372082289, 2.31436414126049041586401831975, 2.60467600555756229149551481858, 2.98004661899199231115437847100, 4.06182673812785905054132006247, 4.44994957570517170152202331319, 5.22465963352202663879592762973, 5.45734437575009036836059962399, 5.71012569325521424791956643699, 6.17106315121314565062659046983, 6.62516017863013169349321112564, 7.47320461628172485062800558626, 7.61068895852562797070369378134, 8.321037332106296729315848288758
