| L(s) = 1 | − 4·3-s + 2·5-s + 6·9-s − 8·15-s − 12·23-s + 3·25-s + 4·27-s − 8·31-s + 20·37-s + 12·45-s + 12·47-s − 2·49-s − 12·53-s − 20·67-s + 48·69-s − 12·75-s − 37·81-s − 12·89-s + 32·93-s − 20·97-s − 4·103-s − 80·111-s + 12·113-s − 24·115-s + 4·125-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 0.894·5-s + 2·9-s − 2.06·15-s − 2.50·23-s + 3/5·25-s + 0.769·27-s − 1.43·31-s + 3.28·37-s + 1.78·45-s + 1.75·47-s − 2/7·49-s − 1.64·53-s − 2.44·67-s + 5.77·69-s − 1.38·75-s − 4.11·81-s − 1.27·89-s + 3.31·93-s − 2.03·97-s − 0.394·103-s − 7.59·111-s + 1.12·113-s − 2.23·115-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93702400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93702400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 134 T^{2} + p^{2} T^{4} \) |
| 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
−7.30257145429964989470127056321, −7.16152133107608755048454354954, −6.46481495347833655471505935935, −6.36681755298627482981543164659, −6.08932151915693439615622831325, −5.77020538277290503662835174098, −5.52003923703056519246466974515, −5.50699551999427299692981136178, −4.65751709201663183287855503289, −4.63751013167091288395795264502, −4.14712762057067364199079420065, −3.92622695030864409994788600927, −2.96320744418115190390988195383, −2.95283724871260692501086256254, −2.22640870311995561967624391743, −1.90207122957652895029283014155, −1.24669399941346977336697117269, −0.911466659384810470525287915962, 0, 0,
0.911466659384810470525287915962, 1.24669399941346977336697117269, 1.90207122957652895029283014155, 2.22640870311995561967624391743, 2.95283724871260692501086256254, 2.96320744418115190390988195383, 3.92622695030864409994788600927, 4.14712762057067364199079420065, 4.63751013167091288395795264502, 4.65751709201663183287855503289, 5.50699551999427299692981136178, 5.52003923703056519246466974515, 5.77020538277290503662835174098, 6.08932151915693439615622831325, 6.36681755298627482981543164659, 6.46481495347833655471505935935, 7.16152133107608755048454354954, 7.30257145429964989470127056321
