L(s) = 1 | + 5-s + 3.66·7-s + 1.21·11-s + 2.21·13-s − 1.21·17-s − 2.57·19-s − 23-s + 25-s − 1.45·29-s − 6.46·31-s + 3.66·35-s + 4·37-s + 10.9·41-s + 6.90·43-s − 5.45·47-s + 6.46·49-s − 3.81·53-s + 1.21·55-s + 4.24·59-s + 6.78·61-s + 2.21·65-s + 12.8·67-s + 6.91·71-s − 15.2·73-s + 4.45·77-s + 15.5·83-s − 1.21·85-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 1.38·7-s + 0.366·11-s + 0.614·13-s − 0.294·17-s − 0.591·19-s − 0.208·23-s + 0.200·25-s − 0.270·29-s − 1.16·31-s + 0.620·35-s + 0.657·37-s + 1.70·41-s + 1.05·43-s − 0.795·47-s + 0.923·49-s − 0.524·53-s + 0.163·55-s + 0.553·59-s + 0.868·61-s + 0.274·65-s + 1.57·67-s + 0.820·71-s − 1.78·73-s + 0.507·77-s + 1.71·83-s − 0.131·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.947100013\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.947100013\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 7 | \( 1 - 3.66T + 7T^{2} \) |
| 11 | \( 1 - 1.21T + 11T^{2} \) |
| 13 | \( 1 - 2.21T + 13T^{2} \) |
| 17 | \( 1 + 1.21T + 17T^{2} \) |
| 19 | \( 1 + 2.57T + 19T^{2} \) |
| 29 | \( 1 + 1.45T + 29T^{2} \) |
| 31 | \( 1 + 6.46T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 - 10.9T + 41T^{2} \) |
| 43 | \( 1 - 6.90T + 43T^{2} \) |
| 47 | \( 1 + 5.45T + 47T^{2} \) |
| 53 | \( 1 + 3.81T + 53T^{2} \) |
| 59 | \( 1 - 4.24T + 59T^{2} \) |
| 61 | \( 1 - 6.78T + 61T^{2} \) |
| 67 | \( 1 - 12.8T + 67T^{2} \) |
| 71 | \( 1 - 6.91T + 71T^{2} \) |
| 73 | \( 1 + 15.2T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 15.5T + 83T^{2} \) |
| 89 | \( 1 - 10.9T + 89T^{2} \) |
| 97 | \( 1 - 1.69T + 97T^{2} \) |
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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
−7.86746385456365779893803671766, −7.19272515199297351046628856603, −6.32310389139123322042350149288, −5.75780409814156535586369535830, −5.00847841059842958600973383944, −4.30387378608267271841401554428, −3.65294056679379818003063604879, −2.40515577100301827776046137610, −1.81300019219809615492812835772, −0.885407525063938122095301999780,
0.885407525063938122095301999780, 1.81300019219809615492812835772, 2.40515577100301827776046137610, 3.65294056679379818003063604879, 4.30387378608267271841401554428, 5.00847841059842958600973383944, 5.75780409814156535586369535830, 6.32310389139123322042350149288, 7.19272515199297351046628856603, 7.86746385456365779893803671766