| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s + 12-s + 16-s − 18-s − 19-s − 24-s − 25-s + 27-s − 32-s + 36-s + 38-s + 2·41-s − 2·43-s + 48-s + 49-s + 50-s − 54-s − 57-s − 2·59-s + 64-s − 72-s − 2·73-s + ⋯ |
| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s + 12-s + 16-s − 18-s − 19-s − 24-s − 25-s + 27-s − 32-s + 36-s + 38-s + 2·41-s − 2·43-s + 48-s + 49-s + 50-s − 54-s − 57-s − 2·59-s + 64-s − 72-s − 2·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7787195592\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7787195592\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + T^{2} \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( ( 1 - T )^{2} \) |
| 43 | \( ( 1 + T )^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 + T )^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 + T )^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 + T )^{2} \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.00397808045101655127229778876, −10.16544002589374197607571660918, −9.399448002762922147301015937186, −8.605456876758143490908913013299, −7.86704251568408462661856995498, −7.02319189845242403376193688009, −5.95802009332349037074083356916, −4.23380588660272063542398513667, −2.94372743923310647162555934661, −1.79130882867276944589858344120,
1.79130882867276944589858344120, 2.94372743923310647162555934661, 4.23380588660272063542398513667, 5.95802009332349037074083356916, 7.02319189845242403376193688009, 7.86704251568408462661856995498, 8.605456876758143490908913013299, 9.399448002762922147301015937186, 10.16544002589374197607571660918, 11.00397808045101655127229778876
