| L(s) = 1 | − 3-s − 5-s + 7-s + 9-s + 5·11-s + 13-s + 15-s + 3·17-s + 6·19-s − 21-s − 7·23-s + 25-s − 27-s − 6·29-s + 10·31-s − 5·33-s − 35-s − 11·37-s − 39-s + 5·41-s + 4·43-s − 45-s + 2·47-s − 6·49-s − 3·51-s − 5·53-s − 5·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 1.50·11-s + 0.277·13-s + 0.258·15-s + 0.727·17-s + 1.37·19-s − 0.218·21-s − 1.45·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s + 1.79·31-s − 0.870·33-s − 0.169·35-s − 1.80·37-s − 0.160·39-s + 0.780·41-s + 0.609·43-s − 0.149·45-s + 0.291·47-s − 6/7·49-s − 0.420·51-s − 0.686·53-s − 0.674·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.678779337\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.678779337\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−8.639433209022781465732453775063, −7.86926379086616910064471941841, −7.21577921922305655926868040451, −6.36449954130063129579960476421, −5.72152571151873638126869526871, −4.82600432045380837460242520937, −3.97042142807256322435240742048, −3.33405399490942012878687906275, −1.78619298567023308744788211654, −0.861696728166987450468023039777,
0.861696728166987450468023039777, 1.78619298567023308744788211654, 3.33405399490942012878687906275, 3.97042142807256322435240742048, 4.82600432045380837460242520937, 5.72152571151873638126869526871, 6.36449954130063129579960476421, 7.21577921922305655926868040451, 7.86926379086616910064471941841, 8.639433209022781465732453775063
