| L(s) = 1 | − i·2-s − 4-s + i·8-s + 3·9-s − 2·11-s − 4i·13-s + 16-s + 6i·17-s − 3i·18-s + 8·19-s + 2i·22-s + 2i·23-s − 4·26-s − 29-s − 31-s − i·32-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + 0.353i·8-s + 9-s − 0.603·11-s − 1.10i·13-s + 0.250·16-s + 1.45i·17-s − 0.707i·18-s + 1.83·19-s + 0.426i·22-s + 0.417i·23-s − 0.784·26-s − 0.185·29-s − 0.179·31-s − 0.176i·32-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
| \(L(1)\) |
\(\approx\) |
\(1.699942627\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.699942627\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 - 8T + 19T^{2} \) |
| 23 | \( 1 - 2iT - 23T^{2} \) |
| 31 | \( 1 + T + 31T^{2} \) |
| 37 | \( 1 + 7iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 9iT - 47T^{2} \) |
| 53 | \( 1 - 12iT - 53T^{2} \) |
| 59 | \( 1 - 15T + 59T^{2} \) |
| 61 | \( 1 - 11T + 61T^{2} \) |
| 67 | \( 1 + 3iT - 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + 12iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 8iT - 83T^{2} \) |
| 89 | \( 1 - 16T + 89T^{2} \) |
| 97 | \( 1 - 2iT - 97T^{2} \) |
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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
−9.585857017823545708884463154615, −8.683370479254036510011848552499, −7.74031506404793690908415237653, −7.23158941478599447701630298295, −5.78514499074730611808600407151, −5.24444080454418888921225294927, −4.03191319487311635399987066311, −3.32860084557082410664160769087, −2.10667647848615049659208939170, −0.912206623876116998074364764163,
1.06542709336287273711984559794, 2.61535697780192640861583366893, 3.86255314370245977803417619261, 4.82185764215319171439433236097, 5.41012565428142092859575305699, 6.67098056811928879251131777832, 7.17793372841849974101854806757, 7.83898261222196686134467131032, 8.874557882551005687964720101931, 9.699428102399344391153195923319
