| L(s) = 1 | − 0.124·2-s − 1.98·4-s + 1.22·5-s + 2.66·7-s + 0.494·8-s − 0.152·10-s + 5.63·11-s − 0.0134·13-s − 0.331·14-s + 3.90·16-s − 3.54·17-s − 5.96·19-s − 2.43·20-s − 0.699·22-s + 4.08·23-s − 3.48·25-s + 0.00167·26-s − 5.29·28-s + 2.96·29-s − 1.47·32-s + 0.440·34-s + 3.27·35-s + 10.6·37-s + 0.740·38-s + 0.608·40-s + 7.14·41-s − 3.17·43-s + ⋯ |
| L(s) = 1 | − 0.0878·2-s − 0.992·4-s + 0.549·5-s + 1.00·7-s + 0.174·8-s − 0.0482·10-s + 1.69·11-s − 0.00373·13-s − 0.0885·14-s + 0.976·16-s − 0.859·17-s − 1.36·19-s − 0.545·20-s − 0.149·22-s + 0.852·23-s − 0.697·25-s + 0.000327·26-s − 1.00·28-s + 0.550·29-s − 0.260·32-s + 0.0754·34-s + 0.554·35-s + 1.74·37-s + 0.120·38-s + 0.0961·40-s + 1.11·41-s − 0.484·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8649 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8649 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.206482686\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.206482686\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 31 | \( 1 \) |
| good | 2 | \( 1 + 0.124T + 2T^{2} \) |
| 5 | \( 1 - 1.22T + 5T^{2} \) |
| 7 | \( 1 - 2.66T + 7T^{2} \) |
| 11 | \( 1 - 5.63T + 11T^{2} \) |
| 13 | \( 1 + 0.0134T + 13T^{2} \) |
| 17 | \( 1 + 3.54T + 17T^{2} \) |
| 19 | \( 1 + 5.96T + 19T^{2} \) |
| 23 | \( 1 - 4.08T + 23T^{2} \) |
| 29 | \( 1 - 2.96T + 29T^{2} \) |
| 37 | \( 1 - 10.6T + 37T^{2} \) |
| 41 | \( 1 - 7.14T + 41T^{2} \) |
| 43 | \( 1 + 3.17T + 43T^{2} \) |
| 47 | \( 1 - 9.81T + 47T^{2} \) |
| 53 | \( 1 + 7.90T + 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 - 3.70T + 61T^{2} \) |
| 67 | \( 1 - 3.56T + 67T^{2} \) |
| 71 | \( 1 + 3.76T + 71T^{2} \) |
| 73 | \( 1 + 3.55T + 73T^{2} \) |
| 79 | \( 1 + 3.25T + 79T^{2} \) |
| 83 | \( 1 + 2.07T + 83T^{2} \) |
| 89 | \( 1 + 8.80T + 89T^{2} \) |
| 97 | \( 1 - 3.38T + 97T^{2} \) |
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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
−7.934529035521204323592134497752, −7.07309197254017743562461697499, −6.27888857598783163278969331119, −5.78619949023977449587576404666, −4.73266102168970595199306083566, −4.37652629585445695994476818625, −3.77671427378659447657939860257, −2.48030832966797027759744736568, −1.62005889590599354771321830846, −0.796488892271977982452317244850,
0.796488892271977982452317244850, 1.62005889590599354771321830846, 2.48030832966797027759744736568, 3.77671427378659447657939860257, 4.37652629585445695994476818625, 4.73266102168970595199306083566, 5.78619949023977449587576404666, 6.27888857598783163278969331119, 7.07309197254017743562461697499, 7.934529035521204323592134497752
