| L(s) = 1 | − i·2-s − 2i·3-s − 4-s − 2·6-s − 2i·7-s + i·8-s − 9-s − 6·11-s + 2i·12-s + 2i·13-s − 2·14-s + 16-s + 6i·17-s + i·18-s − 2·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 1.15i·3-s − 0.5·4-s − 0.816·6-s − 0.755i·7-s + 0.353i·8-s − 0.333·9-s − 1.80·11-s + 0.577i·12-s + 0.554i·13-s − 0.534·14-s + 0.250·16-s + 1.45i·17-s + 0.235i·18-s − 0.458·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 + 2iT - 3T^{2} \) |
| 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 6T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 31 | \( 1 + 7T + 31T^{2} \) |
| 37 | \( 1 - 7iT - 37T^{2} \) |
| 41 | \( 1 + 12T + 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 + 9iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 9T + 59T^{2} \) |
| 61 | \( 1 + T + 61T^{2} \) |
| 67 | \( 1 - iT - 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 4iT - 97T^{2} \) |
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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
−8.601804173317114481841567989198, −8.115711287289803492281037559368, −7.30899608790696544734888275504, −6.56296677293455952721598660339, −5.51672628587367986842256871790, −4.50437246840661578070239812349, −3.47234528854097161946803218723, −2.27091093971533352857860902265, −1.47852189572173631258395816757, 0,
2.45875730195634571050949491854, 3.40470031884530021782360939367, 4.59916385054231776570974859258, 5.29642208402497731274704941599, 5.68196533244516170791628641172, 7.10600051731738631577252129864, 7.73932642641320336328375962718, 8.748367034588930178138098096136, 9.227711834250440085016337198682
