| L(s) = 1 | − 5·7-s + 7·13-s − 2·19-s + 5·25-s + 4·31-s − 2·37-s − 8·43-s + 7·49-s + 13·61-s − 11·67-s + 34·73-s + 13·79-s − 35·91-s − 5·97-s + 7·103-s + 4·109-s + 11·121-s + 127-s + 131-s + 10·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 1.88·7-s + 1.94·13-s − 0.458·19-s + 25-s + 0.718·31-s − 0.328·37-s − 1.21·43-s + 49-s + 1.66·61-s − 1.34·67-s + 3.97·73-s + 1.46·79-s − 3.66·91-s − 0.507·97-s + 0.689·103-s + 0.383·109-s + 121-s + 0.0887·127-s + 0.0873·131-s + 0.867·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.238574621\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.238574621\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
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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
−11.92893567873738895808040613308, −11.12611800332127043043445309373, −11.03604763223208315750936001424, −10.46900672684705922151516453328, −9.800061902910841349221896659816, −9.734468935517001051997820790788, −9.021788410912916157624860181979, −8.499816978974123648859951075744, −8.342785903395719378047634157708, −7.50780876952033811786354442138, −6.64253610396501901993766338709, −6.61546418584568748996332354964, −6.20200305017867850384522648625, −5.52542362782316614216500455019, −4.84678644916868500454409584126, −4.01074531449814063409273287254, −3.40833744756611229016433241825, −3.17112831268645506320487135411, −2.10046681190686105816304517361, −0.825202739181669871320106696033,
0.825202739181669871320106696033, 2.10046681190686105816304517361, 3.17112831268645506320487135411, 3.40833744756611229016433241825, 4.01074531449814063409273287254, 4.84678644916868500454409584126, 5.52542362782316614216500455019, 6.20200305017867850384522648625, 6.61546418584568748996332354964, 6.64253610396501901993766338709, 7.50780876952033811786354442138, 8.342785903395719378047634157708, 8.499816978974123648859951075744, 9.021788410912916157624860181979, 9.734468935517001051997820790788, 9.800061902910841349221896659816, 10.46900672684705922151516453328, 11.03604763223208315750936001424, 11.12611800332127043043445309373, 11.92893567873738895808040613308
