L(s) = 1 | − 2.23i·2-s − i·3-s − 3.00·4-s + (−1 − 2i)5-s − 2.23·6-s + 2.23i·8-s − 9-s + (−4.47 + 2.23i)10-s + 3.00i·12-s − 4.47i·13-s + (−2 + i)15-s − 0.999·16-s + 4.47i·17-s + 2.23i·18-s + (3.00 + 6.00i)20-s + ⋯ |
L(s) = 1 | − 1.58i·2-s − 0.577i·3-s − 1.50·4-s + (−0.447 − 0.894i)5-s − 0.912·6-s + 0.790i·8-s − 0.333·9-s + (−1.41 + 0.707i)10-s + 0.866i·12-s − 1.24i·13-s + (−0.516 + 0.258i)15-s − 0.249·16-s + 1.08i·17-s + 0.527i·18-s + (0.670 + 1.34i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3941379753\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3941379753\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + iT \) |
| 5 | \( 1 + (1 + 2i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 2.23iT - 2T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 13 | \( 1 + 4.47iT - 13T^{2} \) |
| 17 | \( 1 - 4.47iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 + 8.94T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 + 8.94T + 41T^{2} \) |
| 43 | \( 1 - 8.94iT - 43T^{2} \) |
| 47 | \( 1 + 12iT - 47T^{2} \) |
| 53 | \( 1 - 4iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 12iT - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 - 13.4iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 8iT - 97T^{2} \) |
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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
−8.666220609108065011629930803816, −8.012655547793926765874404545616, −7.18566371317146774162636057407, −5.80908696434550465916362382156, −5.13313972688726059222824635259, −3.92203046049116926398319652469, −3.41478905759745497837594884478, −2.13216109295891719054374850049, −1.27889281267816482585358764778, −0.15212646579063749017579757646,
2.37601141727759994178201477679, 3.63651926977594442312264451213, 4.47653462927509338405714176869, 5.23819242570815352755350787687, 6.22975913797124368422794781131, 6.87080100082620743430596886949, 7.40426823121631483002607118915, 8.278226189809698174789571321051, 9.044236019170433713133632072097, 9.673110340659262307730827396729