L(s) = 1 | + (1.36 − 0.366i)2-s + (1.73 − i)4-s + i·5-s + (1.73 − 2i)7-s + (1.99 − 2i)8-s + (0.366 + 1.36i)10-s + 3.73i·11-s + 6.46i·13-s + (1.63 − 3.36i)14-s + (1.99 − 3.46i)16-s − 0.464i·17-s + 6·19-s + (1 + 1.73i)20-s + (1.36 + 5.09i)22-s − 5.46i·23-s + ⋯ |
L(s) = 1 | + (0.965 − 0.258i)2-s + (0.866 − 0.5i)4-s + 0.447i·5-s + (0.654 − 0.755i)7-s + (0.707 − 0.707i)8-s + (0.115 + 0.431i)10-s + 1.12i·11-s + 1.79i·13-s + (0.436 − 0.899i)14-s + (0.499 − 0.866i)16-s − 0.112i·17-s + 1.37·19-s + (0.223 + 0.387i)20-s + (0.291 + 1.08i)22-s − 1.13i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.188i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 + 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.429361646\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.429361646\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.36 + 0.366i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 + (-1.73 + 2i)T \) |
good | 11 | \( 1 - 3.73iT - 11T^{2} \) |
| 13 | \( 1 - 6.46iT - 13T^{2} \) |
| 17 | \( 1 + 0.464iT - 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 + 5.46iT - 23T^{2} \) |
| 29 | \( 1 - 5.92T + 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 + 2.53T + 37T^{2} \) |
| 41 | \( 1 + 3.46iT - 41T^{2} \) |
| 43 | \( 1 + 2iT - 43T^{2} \) |
| 47 | \( 1 - 1.73T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + 3.46T + 59T^{2} \) |
| 61 | \( 1 - 2.53iT - 61T^{2} \) |
| 67 | \( 1 + 3.46iT - 67T^{2} \) |
| 71 | \( 1 - 0.535iT - 71T^{2} \) |
| 73 | \( 1 + 0.928iT - 73T^{2} \) |
| 79 | \( 1 - 2.66iT - 79T^{2} \) |
| 83 | \( 1 + 8.53T + 83T^{2} \) |
| 89 | \( 1 + 9.46iT - 89T^{2} \) |
| 97 | \( 1 - 7.39iT - 97T^{2} \) |
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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
−9.936706138569947183065595767660, −9.011326658471830214300388495903, −7.61042818928212319284520267378, −7.06483076534180305487496772381, −6.46261059134635462931814413017, −5.13837655054351686966944169980, −4.48098876905750623466918200958, −3.73828894406340453306083197308, −2.41376385170474088097405047235, −1.47321954515636268063512811952,
1.30960318687630709106960159714, 2.85196133267664594994675499078, 3.47777774810817988433881205437, 4.85981192687227321082940506024, 5.55871630317663174024237583202, 5.89135157800933352207428538078, 7.31813117714326683715422317539, 8.049654000390553123719221513282, 8.572096755358359822407287188832, 9.718610280449986197389527288507