| L(s) = 1 | − 3-s + 4-s + 9-s − 12-s + 16-s − 2·19-s − 25-s − 27-s + 36-s − 2·37-s − 48-s + 49-s + 2·57-s + 2·61-s + 64-s + 2·67-s − 73-s + 75-s − 2·76-s − 2·79-s + 81-s + 2·97-s − 100-s − 108-s − 2·109-s + 2·111-s + ⋯ |
| L(s) = 1 | − 3-s + 4-s + 9-s − 12-s + 16-s − 2·19-s − 25-s − 27-s + 36-s − 2·37-s − 48-s + 49-s + 2·57-s + 2·61-s + 64-s + 2·67-s − 73-s + 75-s − 2·76-s − 2·79-s + 81-s + 2·97-s − 100-s − 108-s − 2·109-s + 2·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 219 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 219 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6543939537\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6543939537\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 73 | \( 1 + T \) |
| good | 2 | \( ( 1 - T )( 1 + T ) \) |
| 5 | \( 1 + T^{2} \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( ( 1 + T )^{2} \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 + T )^{2} \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( ( 1 - T )^{2} \) |
| 67 | \( ( 1 - T )^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 + T )^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( ( 1 - T )^{2} \) |
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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
−12.32509190118012483117259518174, −11.54511582989197309550654478537, −10.70515094869209504254005589176, −10.04606273348137618322637663066, −8.463713581622620642277735405677, −7.18681845319630877386448672995, −6.41666082408594124643105185568, −5.45192749638810896347645572947, −3.96982476170151320825670643296, −2.02140339971243428967012076641,
2.02140339971243428967012076641, 3.96982476170151320825670643296, 5.45192749638810896347645572947, 6.41666082408594124643105185568, 7.18681845319630877386448672995, 8.463713581622620642277735405677, 10.04606273348137618322637663066, 10.70515094869209504254005589176, 11.54511582989197309550654478537, 12.32509190118012483117259518174
