| L(s) = 1 | + 3·5-s − 9·13-s + 3·17-s + 8·19-s − 9·23-s + 5·25-s − 15·29-s − 8·31-s − 15·41-s − 21·43-s + 3·47-s + 2·49-s − 3·53-s + 3·59-s − 7·61-s − 27·65-s − 5·67-s − 9·71-s − 7·73-s − 7·79-s + 9·85-s − 15·89-s + 24·95-s + 15·97-s + 3·101-s + 32·103-s − 24·107-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 2.49·13-s + 0.727·17-s + 1.83·19-s − 1.87·23-s + 25-s − 2.78·29-s − 1.43·31-s − 2.34·41-s − 3.20·43-s + 0.437·47-s + 2/7·49-s − 0.412·53-s + 0.390·59-s − 0.896·61-s − 3.34·65-s − 0.610·67-s − 1.06·71-s − 0.819·73-s − 0.787·79-s + 0.976·85-s − 1.58·89-s + 2.46·95-s + 1.52·97-s + 0.298·101-s + 3.15·103-s − 2.32·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7901439736\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7901439736\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( 1 - 8 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 9 T + 50 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 15 T + 104 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 15 T + 116 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 3 T + 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 3 T + 56 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 9 T + 10 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 7 T - 30 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 154 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 15 T + 164 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 15 T + 172 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
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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
−9.255139258504715353664231124947, −8.558300385684690624453017341149, −8.397370015652534268373341671914, −7.56706297593131693773460330812, −7.54575132955787889438388616072, −7.17111337240995763810753187466, −6.93938064815398689877059183375, −6.22259444128984158306333876082, −5.65107159353087039085519855755, −5.64095604259584061062358339022, −5.31695715163246919843069387004, −4.70961742426921954335213273304, −4.57542470381745645237484483860, −3.47619937905243329355179866810, −3.45567569001419686576063645668, −2.95523602940204372613899049497, −2.10191905586835977574389162155, −1.74041106135087455574247838880, −1.72999439355718123626895946964, −0.25843238055216035560896507570,
0.25843238055216035560896507570, 1.72999439355718123626895946964, 1.74041106135087455574247838880, 2.10191905586835977574389162155, 2.95523602940204372613899049497, 3.45567569001419686576063645668, 3.47619937905243329355179866810, 4.57542470381745645237484483860, 4.70961742426921954335213273304, 5.31695715163246919843069387004, 5.64095604259584061062358339022, 5.65107159353087039085519855755, 6.22259444128984158306333876082, 6.93938064815398689877059183375, 7.17111337240995763810753187466, 7.54575132955787889438388616072, 7.56706297593131693773460330812, 8.397370015652534268373341671914, 8.558300385684690624453017341149, 9.255139258504715353664231124947
