| L(s) = 1 | − 0.346·3-s − 5-s − 1.09·7-s − 2.88·9-s − 5.77·11-s − 0.497·13-s + 0.346·15-s + 4.15·17-s + 4.48·19-s + 0.378·21-s + 0.346·23-s + 25-s + 2.03·27-s + 29-s + 9.77·31-s + 2·33-s + 1.09·35-s − 2.27·37-s + 0.172·39-s + 0.446·41-s + 5.25·43-s + 2.88·45-s − 2.20·47-s − 5.80·49-s − 1.43·51-s − 13.1·53-s + 5.77·55-s + ⋯ |
| L(s) = 1 | − 0.199·3-s − 0.447·5-s − 0.412·7-s − 0.960·9-s − 1.74·11-s − 0.138·13-s + 0.0893·15-s + 1.00·17-s + 1.02·19-s + 0.0824·21-s + 0.0721·23-s + 0.200·25-s + 0.391·27-s + 0.185·29-s + 1.75·31-s + 0.348·33-s + 0.184·35-s − 0.373·37-s + 0.0275·39-s + 0.0697·41-s + 0.801·43-s + 0.429·45-s − 0.322·47-s − 0.829·49-s − 0.201·51-s − 1.81·53-s + 0.779·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 + 0.346T + 3T^{2} \) |
| 7 | \( 1 + 1.09T + 7T^{2} \) |
| 11 | \( 1 + 5.77T + 11T^{2} \) |
| 13 | \( 1 + 0.497T + 13T^{2} \) |
| 17 | \( 1 - 4.15T + 17T^{2} \) |
| 19 | \( 1 - 4.48T + 19T^{2} \) |
| 23 | \( 1 - 0.346T + 23T^{2} \) |
| 31 | \( 1 - 9.77T + 31T^{2} \) |
| 37 | \( 1 + 2.27T + 37T^{2} \) |
| 41 | \( 1 - 0.446T + 41T^{2} \) |
| 43 | \( 1 - 5.25T + 43T^{2} \) |
| 47 | \( 1 + 2.20T + 47T^{2} \) |
| 53 | \( 1 + 13.1T + 53T^{2} \) |
| 59 | \( 1 + 1.26T + 59T^{2} \) |
| 61 | \( 1 - 3.81T + 61T^{2} \) |
| 67 | \( 1 - 11.1T + 67T^{2} \) |
| 71 | \( 1 - 4.16T + 71T^{2} \) |
| 73 | \( 1 - 13.4T + 73T^{2} \) |
| 79 | \( 1 + 3.22T + 79T^{2} \) |
| 83 | \( 1 - 1.26T + 83T^{2} \) |
| 89 | \( 1 - 13.6T + 89T^{2} \) |
| 97 | \( 1 + 7.09T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.56856782641472125954320272067, −6.63524633705692432344987603911, −5.97046802336497892459425604675, −5.14605440606638217007719555900, −4.93680765839168980268291339571, −3.63514666938774973960760651584, −2.99324867933302348260791508615, −2.48637776559531592409646061457, −0.975348756097127763728427312066, 0,
0.975348756097127763728427312066, 2.48637776559531592409646061457, 2.99324867933302348260791508615, 3.63514666938774973960760651584, 4.93680765839168980268291339571, 5.14605440606638217007719555900, 5.97046802336497892459425604675, 6.63524633705692432344987603911, 7.56856782641472125954320272067
