| L(s) = 1 | − 0.571·2-s − 1.67·4-s + 1.42·7-s + 2.10·8-s − 3.24·11-s + 13-s − 0.816·14-s + 2.14·16-s − 1.85·17-s − 1.81·19-s + 1.85·22-s + 1.52·23-s − 0.571·26-s − 2.38·28-s − 2.34·29-s + 6.38·31-s − 5.42·32-s + 1.06·34-s + 3.52·37-s + 1.03·38-s + 3.81·41-s + 10.0·43-s + 5.42·44-s − 0.874·46-s − 11.2·47-s − 4.96·49-s − 1.67·52-s + ⋯ |
| L(s) = 1 | − 0.404·2-s − 0.836·4-s + 0.539·7-s + 0.742·8-s − 0.978·11-s + 0.277·13-s − 0.218·14-s + 0.535·16-s − 0.450·17-s − 0.416·19-s + 0.395·22-s + 0.318·23-s − 0.112·26-s − 0.451·28-s − 0.435·29-s + 1.14·31-s − 0.959·32-s + 0.182·34-s + 0.580·37-s + 0.168·38-s + 0.596·41-s + 1.52·43-s + 0.818·44-s − 0.128·46-s − 1.64·47-s − 0.708·49-s − 0.231·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.045568482\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.045568482\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + 0.571T + 2T^{2} \) |
| 7 | \( 1 - 1.42T + 7T^{2} \) |
| 11 | \( 1 + 3.24T + 11T^{2} \) |
| 17 | \( 1 + 1.85T + 17T^{2} \) |
| 19 | \( 1 + 1.81T + 19T^{2} \) |
| 23 | \( 1 - 1.52T + 23T^{2} \) |
| 29 | \( 1 + 2.34T + 29T^{2} \) |
| 31 | \( 1 - 6.38T + 31T^{2} \) |
| 37 | \( 1 - 3.52T + 37T^{2} \) |
| 41 | \( 1 - 3.81T + 41T^{2} \) |
| 43 | \( 1 - 10.0T + 43T^{2} \) |
| 47 | \( 1 + 11.2T + 47T^{2} \) |
| 53 | \( 1 + 6.81T + 53T^{2} \) |
| 59 | \( 1 - 5.91T + 59T^{2} \) |
| 61 | \( 1 - 5.48T + 61T^{2} \) |
| 67 | \( 1 + 10.3T + 67T^{2} \) |
| 71 | \( 1 - 9.81T + 71T^{2} \) |
| 73 | \( 1 - 5.32T + 73T^{2} \) |
| 79 | \( 1 + 2.96T + 79T^{2} \) |
| 83 | \( 1 + 3.14T + 83T^{2} \) |
| 89 | \( 1 - 4.85T + 89T^{2} \) |
| 97 | \( 1 - 1.51T + 97T^{2} \) |
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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
−8.668697616250701518990440000162, −8.084573216302596633423234408745, −7.58396251758160208117676833557, −6.50425382825347131330507456218, −5.57546833097082822369003940830, −4.79046465287636534187215303353, −4.23264518193725686854590999728, −3.07134741756258355943753264437, −1.93479004128584152900525663201, −0.67492857874233498864615633360,
0.67492857874233498864615633360, 1.93479004128584152900525663201, 3.07134741756258355943753264437, 4.23264518193725686854590999728, 4.79046465287636534187215303353, 5.57546833097082822369003940830, 6.50425382825347131330507456218, 7.58396251758160208117676833557, 8.084573216302596633423234408745, 8.668697616250701518990440000162
