| L(s) = 1 | − 0.704·3-s + 5-s + 7-s − 2.50·9-s + 1.86·11-s + 13-s − 0.704·15-s + 0.491·17-s − 1.86·19-s − 0.704·21-s + 3.76·23-s + 25-s + 3.87·27-s + 8.13·29-s − 5.24·31-s − 1.31·33-s + 35-s − 3.87·37-s − 0.704·39-s − 5.85·41-s + 6.02·43-s − 2.50·45-s − 3.69·47-s + 49-s − 0.345·51-s − 6.27·53-s + 1.86·55-s + ⋯ |
| L(s) = 1 | − 0.406·3-s + 0.447·5-s + 0.377·7-s − 0.834·9-s + 0.562·11-s + 0.277·13-s − 0.181·15-s + 0.119·17-s − 0.427·19-s − 0.153·21-s + 0.785·23-s + 0.200·25-s + 0.745·27-s + 1.51·29-s − 0.942·31-s − 0.228·33-s + 0.169·35-s − 0.636·37-s − 0.112·39-s − 0.914·41-s + 0.919·43-s − 0.373·45-s − 0.538·47-s + 0.142·49-s − 0.0484·51-s − 0.862·53-s + 0.251·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.777877520\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.777877520\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 0.704T + 3T^{2} \) |
| 11 | \( 1 - 1.86T + 11T^{2} \) |
| 17 | \( 1 - 0.491T + 17T^{2} \) |
| 19 | \( 1 + 1.86T + 19T^{2} \) |
| 23 | \( 1 - 3.76T + 23T^{2} \) |
| 29 | \( 1 - 8.13T + 29T^{2} \) |
| 31 | \( 1 + 5.24T + 31T^{2} \) |
| 37 | \( 1 + 3.87T + 37T^{2} \) |
| 41 | \( 1 + 5.85T + 41T^{2} \) |
| 43 | \( 1 - 6.02T + 43T^{2} \) |
| 47 | \( 1 + 3.69T + 47T^{2} \) |
| 53 | \( 1 + 6.27T + 53T^{2} \) |
| 59 | \( 1 + 1.51T + 59T^{2} \) |
| 61 | \( 1 - 11.2T + 61T^{2} \) |
| 67 | \( 1 - 11.2T + 67T^{2} \) |
| 71 | \( 1 - 8.55T + 71T^{2} \) |
| 73 | \( 1 - 9.94T + 73T^{2} \) |
| 79 | \( 1 - 8.08T + 79T^{2} \) |
| 83 | \( 1 - 5.46T + 83T^{2} \) |
| 89 | \( 1 + 12.6T + 89T^{2} \) |
| 97 | \( 1 + 0.919T + 97T^{2} \) |
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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
−8.570995549800456320355915386257, −7.946618091548126911954415914036, −6.74582647117781879263100514003, −6.45180193723324399889093037550, −5.41719268783668322302838674377, −4.99923810886566292604999206583, −3.89955322416481755427954904314, −2.97395837243508311219913713587, −1.95299715743010607300542189140, −0.809767183342734418819452825773,
0.809767183342734418819452825773, 1.95299715743010607300542189140, 2.97395837243508311219913713587, 3.89955322416481755427954904314, 4.99923810886566292604999206583, 5.41719268783668322302838674377, 6.45180193723324399889093037550, 6.74582647117781879263100514003, 7.946618091548126911954415914036, 8.570995549800456320355915386257
