| L(s) = 1 | + 2.82·5-s − 7-s − 3·9-s + 2.82·11-s − 2.82·13-s − 2·17-s + 5.65·19-s + 8·23-s + 3.00·25-s − 5.65·29-s + 8·31-s − 2.82·35-s + 5.65·37-s + 6·41-s + 2.82·43-s − 8.48·45-s + 8·47-s + 49-s + 11.3·53-s + 8.00·55-s − 11.3·59-s + 2.82·61-s + 3·63-s − 8.00·65-s − 8.48·67-s + 8·71-s − 6·73-s + ⋯ |
| L(s) = 1 | + 1.26·5-s − 0.377·7-s − 9-s + 0.852·11-s − 0.784·13-s − 0.485·17-s + 1.29·19-s + 1.66·23-s + 0.600·25-s − 1.05·29-s + 1.43·31-s − 0.478·35-s + 0.929·37-s + 0.937·41-s + 0.431·43-s − 1.26·45-s + 1.16·47-s + 0.142·49-s + 1.55·53-s + 1.07·55-s − 1.47·59-s + 0.362·61-s + 0.377·63-s − 0.992·65-s − 1.03·67-s + 0.949·71-s − 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.070289289\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.070289289\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 + 3T^{2} \) |
| 5 | \( 1 - 2.82T + 5T^{2} \) |
| 11 | \( 1 - 2.82T + 11T^{2} \) |
| 13 | \( 1 + 2.82T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 5.65T + 19T^{2} \) |
| 23 | \( 1 - 8T + 23T^{2} \) |
| 29 | \( 1 + 5.65T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 - 5.65T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 2.82T + 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 - 11.3T + 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 - 2.82T + 61T^{2} \) |
| 67 | \( 1 + 8.48T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 5.65T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 14T + 97T^{2} \) |
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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
−9.283207291184141466584345986319, −8.849715706945957713738810475150, −7.58342662965889560029930902122, −6.82202198869697479163613700476, −5.94105328764709052778105350747, −5.45851803601920151136071006506, −4.41284954827906460242568726079, −3.06394302889213970363524277298, −2.41233067412755602077374319604, −1.02142960226701437714894546397,
1.02142960226701437714894546397, 2.41233067412755602077374319604, 3.06394302889213970363524277298, 4.41284954827906460242568726079, 5.45851803601920151136071006506, 5.94105328764709052778105350747, 6.82202198869697479163613700476, 7.58342662965889560029930902122, 8.849715706945957713738810475150, 9.283207291184141466584345986319
