| L(s) = 1 | − 5-s + 0.732·7-s + 6.19·11-s − 1.26·13-s + 7.19·17-s + 5.73·19-s − 6.46·23-s + 25-s − 7.66·29-s + 1.19·31-s − 0.732·35-s − 0.535·37-s + 6.73·41-s + 6.73·43-s − 0.535·47-s − 6.46·49-s + 13.1·53-s − 6.19·55-s − 1.26·59-s − 1.92·61-s + 1.26·65-s − 12.9·67-s + 11.6·71-s − 10.1·73-s + 4.53·77-s − 10.1·79-s − 12.8·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.276·7-s + 1.86·11-s − 0.351·13-s + 1.74·17-s + 1.31·19-s − 1.34·23-s + 0.200·25-s − 1.42·29-s + 0.214·31-s − 0.123·35-s − 0.0881·37-s + 1.05·41-s + 1.02·43-s − 0.0781·47-s − 0.923·49-s + 1.81·53-s − 0.835·55-s − 0.165·59-s − 0.246·61-s + 0.157·65-s − 1.57·67-s + 1.38·71-s − 1.19·73-s + 0.516·77-s − 1.13·79-s − 1.41·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.202450197\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.202450197\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| good | 7 | \( 1 - 0.732T + 7T^{2} \) |
| 11 | \( 1 - 6.19T + 11T^{2} \) |
| 13 | \( 1 + 1.26T + 13T^{2} \) |
| 17 | \( 1 - 7.19T + 17T^{2} \) |
| 19 | \( 1 - 5.73T + 19T^{2} \) |
| 23 | \( 1 + 6.46T + 23T^{2} \) |
| 29 | \( 1 + 7.66T + 29T^{2} \) |
| 31 | \( 1 - 1.19T + 31T^{2} \) |
| 37 | \( 1 + 0.535T + 37T^{2} \) |
| 41 | \( 1 - 6.73T + 41T^{2} \) |
| 43 | \( 1 - 6.73T + 43T^{2} \) |
| 47 | \( 1 + 0.535T + 47T^{2} \) |
| 53 | \( 1 - 13.1T + 53T^{2} \) |
| 59 | \( 1 + 1.26T + 59T^{2} \) |
| 61 | \( 1 + 1.92T + 61T^{2} \) |
| 67 | \( 1 + 12.9T + 67T^{2} \) |
| 71 | \( 1 - 11.6T + 71T^{2} \) |
| 73 | \( 1 + 10.1T + 73T^{2} \) |
| 79 | \( 1 + 10.1T + 79T^{2} \) |
| 83 | \( 1 + 12.8T + 83T^{2} \) |
| 89 | \( 1 - 3.80T + 89T^{2} \) |
| 97 | \( 1 + 5.46T + 97T^{2} \) |
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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
−8.313411948504778140037771297331, −7.48815167854089382611655641566, −7.21974591012469758035604950848, −5.99579134842451934810639287726, −5.62096192627481766165189793351, −4.45475829255172912710422734195, −3.81899516224035018542172005839, −3.13145604555847322077398472762, −1.76007827027691236860390905892, −0.900919006441812386359066305610,
0.900919006441812386359066305610, 1.76007827027691236860390905892, 3.13145604555847322077398472762, 3.81899516224035018542172005839, 4.45475829255172912710422734195, 5.62096192627481766165189793351, 5.99579134842451934810639287726, 7.21974591012469758035604950848, 7.48815167854089382611655641566, 8.313411948504778140037771297331
