L(s) = 1 | + i·2-s + 3-s − 4-s − i·5-s + i·6-s − 2i·7-s − i·8-s + 9-s + 10-s − 12-s + 2·14-s − i·15-s + 16-s + i·18-s + 2i·19-s + i·20-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + 0.577·3-s − 0.5·4-s − 0.447i·5-s + 0.408i·6-s − 0.755i·7-s − 0.353i·8-s + 0.333·9-s + 0.316·10-s − 0.288·12-s + 0.534·14-s − 0.258i·15-s + 0.250·16-s + 0.235i·18-s + 0.458i·19-s + 0.223i·20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.097657410\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.097657410\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + iT \) |
| 13 | \( 1 \) |
good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 2iT - 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 4iT - 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 6iT - 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 8iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 + 6iT - 89T^{2} \) |
| 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
−7.957909397047282314732791807547, −7.56203604562913967617033268408, −6.88481688968352137943313621485, −6.05688339399078210721993952036, −5.24331532351623691715885429779, −4.44398402993999719145757551735, −3.83875701704384749159726300700, −2.92301842864586702579521695990, −1.66860415195555780406686943718, −0.57548045762163227066885860546,
1.12664987648233769261960092703, 2.21395341874504585781092200842, 2.89190716999261922645646820403, 3.48082916968044289119079336935, 4.55224464269995044246134600256, 5.18371594968828039866394849513, 6.15564634257488720778145523548, 6.91327415804241022931897801510, 7.72396099817686158039038024753, 8.453759769084378217737790441724