| L(s) = 1 | − 2·5-s − 7-s + 6·13-s + 2·17-s − 4·19-s − 4·23-s − 25-s + 10·29-s + 8·31-s + 2·35-s + 6·37-s + 2·41-s + 4·43-s + 8·47-s + 49-s + 10·53-s + 12·59-s − 2·61-s − 12·65-s − 12·67-s − 12·71-s − 14·73-s + 8·79-s + 12·83-s − 4·85-s + 2·89-s − 6·91-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.377·7-s + 1.66·13-s + 0.485·17-s − 0.917·19-s − 0.834·23-s − 1/5·25-s + 1.85·29-s + 1.43·31-s + 0.338·35-s + 0.986·37-s + 0.312·41-s + 0.609·43-s + 1.16·47-s + 1/7·49-s + 1.37·53-s + 1.56·59-s − 0.256·61-s − 1.48·65-s − 1.46·67-s − 1.42·71-s − 1.63·73-s + 0.900·79-s + 1.31·83-s − 0.433·85-s + 0.211·89-s − 0.628·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.340324021\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.340324021\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−10.18277858195260033340705124604, −8.913562206930638585716012763276, −8.334199259994987818549291908705, −7.60026307977982948167798436547, −6.43798012578301106351522548543, −5.90970829345295627546285462325, −4.40034147894441213970326651837, −3.82931598942638938100602509659, −2.67493504075964250922211563542, −0.925974150930484580155748442830,
0.925974150930484580155748442830, 2.67493504075964250922211563542, 3.82931598942638938100602509659, 4.40034147894441213970326651837, 5.90970829345295627546285462325, 6.43798012578301106351522548543, 7.60026307977982948167798436547, 8.334199259994987818549291908705, 8.913562206930638585716012763276, 10.18277858195260033340705124604
