| L(s) = 1 | + i·3-s + (−2 − i)5-s − i·7-s + 2·9-s + 11-s − i·13-s + (1 − 2i)15-s − 3i·17-s − 4·19-s + 21-s − 2i·23-s + (3 + 4i)25-s + 5i·27-s − 29-s − 6·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + (−0.894 − 0.447i)5-s − 0.377i·7-s + 0.666·9-s + 0.301·11-s − 0.277i·13-s + (0.258 − 0.516i)15-s − 0.727i·17-s − 0.917·19-s + 0.218·21-s − 0.417i·23-s + (0.600 + 0.800i)25-s + 0.962i·27-s − 0.185·29-s − 1.07·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7546741021\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7546741021\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (2 + i)T \) |
| 7 | \( 1 + iT \) |
| good | 3 | \( 1 - iT - 3T^{2} \) |
| 11 | \( 1 - T + 11T^{2} \) |
| 13 | \( 1 + iT - 13T^{2} \) |
| 17 | \( 1 + 3iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + 2iT - 23T^{2} \) |
| 29 | \( 1 + T + 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 - 9iT - 47T^{2} \) |
| 53 | \( 1 + 14iT - 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 - 4T + 61T^{2} \) |
| 67 | \( 1 + 10iT - 67T^{2} \) |
| 71 | \( 1 + 16T + 71T^{2} \) |
| 73 | \( 1 + 10iT - 73T^{2} \) |
| 79 | \( 1 - 11T + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 + 12T + 89T^{2} \) |
| 97 | \( 1 + 19iT - 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.817657413218112451353241511906, −8.058417322185325920710096687510, −7.24982985865202971765063356171, −6.64417896302252110208382440785, −5.35621408071497572020905633756, −4.62272669291462483041148488468, −3.98573084281879101764207324267, −3.22355143435925611147062859124, −1.67886796995363066051158480688, −0.26782730409805383087605093350,
1.41627846007304574185549093111, 2.43008637385855123559522439640, 3.69132277762757181282166305959, 4.22736966864715661688539858549, 5.39078557897195606851427934222, 6.45597334346965511796262917175, 6.93655542602082511837804887232, 7.68793383710606547363503198831, 8.417383794736578545521292719953, 9.093539465799854060103428884620
