| L(s) = 1 | − 1.03i·5-s − i·7-s − 0.378·11-s + 2·13-s − 3.86i·17-s − 1.46i·19-s − 5.27·23-s + 3.92·25-s − 3.48i·29-s − 2.53i·31-s − 1.03·35-s + 2.92·37-s − 8.76i·41-s − 4i·43-s − 8.48·47-s + ⋯ |
| L(s) = 1 | − 0.462i·5-s − 0.377i·7-s − 0.114·11-s + 0.554·13-s − 0.937i·17-s − 0.335i·19-s − 1.10·23-s + 0.785·25-s − 0.647i·29-s − 0.455i·31-s − 0.174·35-s + 0.481·37-s − 1.36i·41-s − 0.609i·43-s − 1.23·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0917 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0917 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.387828869\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.387828869\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + iT \) |
| good | 5 | \( 1 + 1.03iT - 5T^{2} \) |
| 11 | \( 1 + 0.378T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 3.86iT - 17T^{2} \) |
| 19 | \( 1 + 1.46iT - 19T^{2} \) |
| 23 | \( 1 + 5.27T + 23T^{2} \) |
| 29 | \( 1 + 3.48iT - 29T^{2} \) |
| 31 | \( 1 + 2.53iT - 31T^{2} \) |
| 37 | \( 1 - 2.92T + 37T^{2} \) |
| 41 | \( 1 + 8.76iT - 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 8.48T + 47T^{2} \) |
| 53 | \( 1 + 7.07iT - 53T^{2} \) |
| 59 | \( 1 - 2.82T + 59T^{2} \) |
| 61 | \( 1 - 3.46T + 61T^{2} \) |
| 67 | \( 1 - 8.53iT - 67T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 - 7.46T + 73T^{2} \) |
| 79 | \( 1 + 10.3iT - 79T^{2} \) |
| 83 | \( 1 - 17.5T + 83T^{2} \) |
| 89 | \( 1 - 1.03iT - 89T^{2} \) |
| 97 | \( 1 - 4.53T + 97T^{2} \) |
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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
−9.757229400452677877082079813268, −8.935681440107548027273065357419, −8.148297824668986008753239644382, −7.29047316638541292002026292313, −6.37988067644478285206318520448, −5.38312563394367346571900622658, −4.49120969035020639278749544037, −3.51273057835959194785967456982, −2.18389954776652517743319820053, −0.65674290018452431764736875285,
1.55737198546643798236416322612, 2.86468064904518288378437143387, 3.83338147656519739423359666976, 4.96114742878609849501662085608, 6.06631664206372537542708065040, 6.59010114563411695614590210980, 7.80954169281010732069370228657, 8.407469547965208688137000608435, 9.352208202651053521922555538458, 10.24844848908474358133174203870
