| L(s) = 1 | + 2.41·2-s + 3.82·4-s + 5-s − 0.585·7-s + 4.41·8-s + 2.41·10-s + 2.82·11-s − 2.58·13-s − 1.41·14-s + 2.99·16-s + 4·17-s + 2.82·19-s + 3.82·20-s + 6.82·22-s + 6·23-s + 25-s − 6.24·26-s − 2.24·28-s − 2.24·29-s + 31-s − 1.58·32-s + 9.65·34-s − 0.585·35-s − 1.41·37-s + 6.82·38-s + 4.41·40-s + 0.828·41-s + ⋯ |
| L(s) = 1 | + 1.70·2-s + 1.91·4-s + 0.447·5-s − 0.221·7-s + 1.56·8-s + 0.763·10-s + 0.852·11-s − 0.717·13-s − 0.377·14-s + 0.749·16-s + 0.970·17-s + 0.648·19-s + 0.856·20-s + 1.45·22-s + 1.25·23-s + 0.200·25-s − 1.22·26-s − 0.423·28-s − 0.416·29-s + 0.179·31-s − 0.280·32-s + 1.65·34-s − 0.0990·35-s − 0.232·37-s + 1.10·38-s + 0.697·40-s + 0.129·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1395 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1395 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.005597631\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.005597631\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 - 2.41T + 2T^{2} \) |
| 7 | \( 1 + 0.585T + 7T^{2} \) |
| 11 | \( 1 - 2.82T + 11T^{2} \) |
| 13 | \( 1 + 2.58T + 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 + 2.24T + 29T^{2} \) |
| 37 | \( 1 + 1.41T + 37T^{2} \) |
| 41 | \( 1 - 0.828T + 41T^{2} \) |
| 43 | \( 1 + 11.3T + 43T^{2} \) |
| 47 | \( 1 - 4.82T + 47T^{2} \) |
| 53 | \( 1 - 4T + 53T^{2} \) |
| 59 | \( 1 - 0.242T + 59T^{2} \) |
| 61 | \( 1 + 10.4T + 61T^{2} \) |
| 67 | \( 1 - 3.89T + 67T^{2} \) |
| 71 | \( 1 + 9.89T + 71T^{2} \) |
| 73 | \( 1 + 5.89T + 73T^{2} \) |
| 79 | \( 1 + 14.4T + 79T^{2} \) |
| 83 | \( 1 - 0.343T + 83T^{2} \) |
| 89 | \( 1 + 5.07T + 89T^{2} \) |
| 97 | \( 1 - 15.6T + 97T^{2} \) |
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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
−9.689776917751776010507135608467, −8.860508155884864901772656986489, −7.49315527707448723633234249661, −6.87246951693514748816529465738, −6.03681737380759492830170957249, −5.29455937310970767197650876019, −4.59428845373093732108033239652, −3.49774407155196077313987631921, −2.84999261606262870939630256300, −1.53443582085226265265461819891,
1.53443582085226265265461819891, 2.84999261606262870939630256300, 3.49774407155196077313987631921, 4.59428845373093732108033239652, 5.29455937310970767197650876019, 6.03681737380759492830170957249, 6.87246951693514748816529465738, 7.49315527707448723633234249661, 8.860508155884864901772656986489, 9.689776917751776010507135608467
