L(s) = 1 | + (−1.30 − 0.541i)2-s + (−1.13 + 1.30i)3-s + (1.41 + 1.41i)4-s + (2.19 − 1.09i)6-s + 2.27i·7-s + (−1.08 − 2.61i)8-s + (−0.414 − 2.97i)9-s − 4.20i·11-s + (−3.45 + 0.239i)12-s − 3.21i·13-s + (1.23 − 2.97i)14-s + 4i·16-s − 1.53i·17-s + (−1.06 + 4.10i)18-s + 4.82·19-s + ⋯ |
L(s) = 1 | + (−0.923 − 0.382i)2-s + (−0.656 + 0.754i)3-s + (0.707 + 0.707i)4-s + (0.895 − 0.445i)6-s + 0.859i·7-s + (−0.382 − 0.923i)8-s + (−0.138 − 0.990i)9-s − 1.26i·11-s + (−0.997 + 0.0692i)12-s − 0.891i·13-s + (0.328 − 0.794i)14-s + i·16-s − 0.371i·17-s + (−0.251 + 0.967i)18-s + 1.10·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.317i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 + 0.317i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.716666 - 0.116921i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.716666 - 0.116921i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.30 + 0.541i)T \) |
| 3 | \( 1 + (1.13 - 1.30i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 2.27iT - 7T^{2} \) |
| 11 | \( 1 + 4.20iT - 11T^{2} \) |
| 13 | \( 1 + 3.21iT - 13T^{2} \) |
| 17 | \( 1 + 1.53iT - 17T^{2} \) |
| 19 | \( 1 - 4.82T + 19T^{2} \) |
| 23 | \( 1 - 1.08T + 23T^{2} \) |
| 29 | \( 1 + 1.74T + 29T^{2} \) |
| 31 | \( 1 - 6.82iT - 31T^{2} \) |
| 37 | \( 1 + 7.76iT - 37T^{2} \) |
| 41 | \( 1 - 2.46iT - 41T^{2} \) |
| 43 | \( 1 - 8.70T + 43T^{2} \) |
| 47 | \( 1 - 1.08T + 47T^{2} \) |
| 53 | \( 1 - 11.0T + 53T^{2} \) |
| 59 | \( 1 + 4.20iT - 59T^{2} \) |
| 61 | \( 1 + 8.48iT - 61T^{2} \) |
| 67 | \( 1 + 2.27T + 67T^{2} \) |
| 71 | \( 1 - 11.8T + 71T^{2} \) |
| 73 | \( 1 + 4.54T + 73T^{2} \) |
| 79 | \( 1 + 0.485iT - 79T^{2} \) |
| 83 | \( 1 - 6.94iT - 83T^{2} \) |
| 89 | \( 1 - 8.40iT - 89T^{2} \) |
| 97 | \( 1 - 10.9T + 97T^{2} \) |
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
−10.74567488000869351349808355643, −9.708555744857046610217160120779, −9.048004564819737982463910946443, −8.309195118097609798611083643531, −7.15311763957571640347679330641, −5.93767416507132191538607016336, −5.32229206710586742132268660029, −3.64417222800599961821873581140, −2.79817797195833232287349435283, −0.75180331196397507009550300455,
1.08715362494179737449356831718, 2.25402484736388231680115844041, 4.34007695181345818954566452380, 5.48837206411702211695173045020, 6.52894808088538195948231591725, 7.30895186171783329507566024329, 7.66830792567822224457963657453, 8.967925976712835988252910148915, 9.914246391945465404993038654994, 10.56314829865302093803279892264