| L(s) = 1 | − 3-s − 2·5-s + 4·7-s + 9-s − 2·13-s + 2·15-s − 17-s + 8·19-s − 4·21-s + 6·23-s − 25-s − 27-s − 2·29-s + 4·31-s − 8·35-s + 6·37-s + 2·39-s − 6·41-s + 4·43-s − 2·45-s + 12·47-s + 9·49-s + 51-s − 8·57-s − 6·59-s − 12·61-s + 4·63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1.51·7-s + 1/3·9-s − 0.554·13-s + 0.516·15-s − 0.242·17-s + 1.83·19-s − 0.872·21-s + 1.25·23-s − 1/5·25-s − 0.192·27-s − 0.371·29-s + 0.718·31-s − 1.35·35-s + 0.986·37-s + 0.320·39-s − 0.937·41-s + 0.609·43-s − 0.298·45-s + 1.75·47-s + 9/7·49-s + 0.140·51-s − 1.05·57-s − 0.781·59-s − 1.53·61-s + 0.503·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.422143733\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.422143733\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.68861779394940, −13.47303089118454, −12.50196901852671, −12.19801290509385, −11.72603241382482, −11.39438995762797, −10.96744164694695, −10.55104590848546, −9.831849457048590, −9.243328666827439, −8.856484816105479, −7.934544692603834, −7.820846040174275, −7.354974487349461, −6.850922184245809, −6.044958918628762, −5.416243913696604, −4.986681394399924, −4.588328427739178, −4.029763953712793, −3.309041030914029, −2.649538226065880, −1.847114047907338, −1.088054216380447, −0.6036304735793320,
0.6036304735793320, 1.088054216380447, 1.847114047907338, 2.649538226065880, 3.309041030914029, 4.029763953712793, 4.588328427739178, 4.986681394399924, 5.416243913696604, 6.044958918628762, 6.850922184245809, 7.354974487349461, 7.820846040174275, 7.934544692603834, 8.856484816105479, 9.243328666827439, 9.831849457048590, 10.55104590848546, 10.96744164694695, 11.39438995762797, 11.72603241382482, 12.19801290509385, 12.50196901852671, 13.47303089118454, 13.68861779394940
