| L(s) = 1 | − 3-s + 5-s − 7-s + 11-s − 13-s − 15-s + 17-s + 21-s + 25-s + 27-s + 29-s − 33-s − 35-s + 39-s + 47-s + 49-s − 51-s + 55-s − 65-s + 2·71-s − 2·73-s − 75-s − 77-s − 79-s − 81-s + 2·83-s + 85-s + ⋯ |
| L(s) = 1 | − 3-s + 5-s − 7-s + 11-s − 13-s − 15-s + 17-s + 21-s + 25-s + 27-s + 29-s − 33-s − 35-s + 39-s + 47-s + 49-s − 51-s + 55-s − 65-s + 2·71-s − 2·73-s − 75-s − 77-s − 79-s − 81-s + 2·83-s + 85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9011659481\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9011659481\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 + T + T^{2} \) |
| 11 | \( 1 - T + T^{2} \) |
| 13 | \( 1 + T + T^{2} \) |
| 17 | \( 1 - T + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( 1 - T + T^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( 1 - T + T^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )^{2} \) |
| 73 | \( ( 1 + T )^{2} \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( ( 1 - T )^{2} \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( 1 - T + T^{2} \) |
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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
−9.374604998051944660209708983297, −8.692016180832189638682762171679, −7.40643534041186724801619332297, −6.62220552071558489495359773234, −6.11423946741065666326612462640, −5.44977311974315207852571759916, −4.63879820689454096748119109095, −3.38839911512926316428060272328, −2.43473540771854653701908477771, −0.982940633817558349115828193711,
0.982940633817558349115828193711, 2.43473540771854653701908477771, 3.38839911512926316428060272328, 4.63879820689454096748119109095, 5.44977311974315207852571759916, 6.11423946741065666326612462640, 6.62220552071558489495359773234, 7.40643534041186724801619332297, 8.692016180832189638682762171679, 9.374604998051944660209708983297
