| L(s) = 1 | + 2·3-s − 4·5-s + 3·9-s − 8·15-s + 4·17-s + 4·19-s + 2·25-s + 4·27-s − 16·31-s − 12·45-s − 14·49-s + 8·51-s + 8·57-s − 8·59-s − 4·61-s + 8·67-s − 16·71-s + 20·73-s + 4·75-s + 16·79-s + 5·81-s − 16·85-s − 32·93-s − 16·95-s − 36·101-s − 32·103-s + 24·107-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.78·5-s + 9-s − 2.06·15-s + 0.970·17-s + 0.917·19-s + 2/5·25-s + 0.769·27-s − 2.87·31-s − 1.78·45-s − 2·49-s + 1.12·51-s + 1.05·57-s − 1.04·59-s − 0.512·61-s + 0.977·67-s − 1.89·71-s + 2.34·73-s + 0.461·75-s + 1.80·79-s + 5/9·81-s − 1.73·85-s − 3.31·93-s − 1.64·95-s − 3.58·101-s − 3.15·103-s + 2.32·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 415872 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 415872 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 19 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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
−8.173282349834764667604075089133, −7.979156449589182873646432658217, −7.61950951666595077126724317917, −7.26958585488936859387169650645, −6.85891827919805546068039331531, −6.09791694801381232421198374318, −5.40997773154525098146238432632, −5.01357758335939619070346679818, −4.23309352016035067675133294948, −3.83764998170958613523904255647, −3.37632188765103348554746581671, −3.14826068230388029805668446658, −2.12120802782336302555457408398, −1.36402928363394079803758664889, 0,
1.36402928363394079803758664889, 2.12120802782336302555457408398, 3.14826068230388029805668446658, 3.37632188765103348554746581671, 3.83764998170958613523904255647, 4.23309352016035067675133294948, 5.01357758335939619070346679818, 5.40997773154525098146238432632, 6.09791694801381232421198374318, 6.85891827919805546068039331531, 7.26958585488936859387169650645, 7.61950951666595077126724317917, 7.979156449589182873646432658217, 8.173282349834764667604075089133
