| L(s) = 1 | + 3-s − 5-s + 7-s + 9-s + 4·11-s − 2·13-s − 15-s − 6·17-s + 21-s + 8·23-s + 25-s + 27-s + 10·29-s + 8·31-s + 4·33-s − 35-s + 2·37-s − 2·39-s − 2·41-s − 8·43-s − 45-s − 4·47-s + 49-s − 6·51-s + 10·53-s − 4·55-s − 4·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 1.20·11-s − 0.554·13-s − 0.258·15-s − 1.45·17-s + 0.218·21-s + 1.66·23-s + 1/5·25-s + 0.192·27-s + 1.85·29-s + 1.43·31-s + 0.696·33-s − 0.169·35-s + 0.328·37-s − 0.320·39-s − 0.312·41-s − 1.21·43-s − 0.149·45-s − 0.583·47-s + 1/7·49-s − 0.840·51-s + 1.37·53-s − 0.539·55-s − 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.171955227\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.171955227\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−9.115112598794557806117561598894, −8.657145943398661951778899302420, −7.897549533550666915091108893756, −6.80988996049369149411313055675, −6.54115760860854978934135332179, −4.87389466393525573091079927707, −4.46294950855221479583418500707, −3.34111414669508805550455957055, −2.38494539489220326364029157676, −1.05635901521155501545827828475,
1.05635901521155501545827828475, 2.38494539489220326364029157676, 3.34111414669508805550455957055, 4.46294950855221479583418500707, 4.87389466393525573091079927707, 6.54115760860854978934135332179, 6.80988996049369149411313055675, 7.897549533550666915091108893756, 8.657145943398661951778899302420, 9.115112598794557806117561598894
