L(s) = 1 | − i·2-s − i·3-s − 4-s + (−0.594 + 2.15i)5-s − 6-s + 1.17i·7-s + i·8-s − 9-s + (2.15 + 0.594i)10-s − 0.822·11-s + i·12-s − 1.74i·13-s + 1.17·14-s + (2.15 + 0.594i)15-s + 16-s − 3.94i·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.577i·3-s − 0.5·4-s + (−0.265 + 0.964i)5-s − 0.408·6-s + 0.445i·7-s + 0.353i·8-s − 0.333·9-s + (0.681 + 0.187i)10-s − 0.247·11-s + 0.288i·12-s − 0.485i·13-s + 0.314·14-s + (0.556 + 0.153i)15-s + 0.250·16-s − 0.956i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 - 0.265i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4229635403\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4229635403\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + iT \) |
| 5 | \( 1 + (0.594 - 2.15i)T \) |
| 37 | \( 1 - iT \) |
good | 7 | \( 1 - 1.17iT - 7T^{2} \) |
| 11 | \( 1 + 0.822T + 11T^{2} \) |
| 13 | \( 1 + 1.74iT - 13T^{2} \) |
| 17 | \( 1 + 3.94iT - 17T^{2} \) |
| 19 | \( 1 + 1.29T + 19T^{2} \) |
| 23 | \( 1 + 7.68iT - 23T^{2} \) |
| 29 | \( 1 + 9.23T + 29T^{2} \) |
| 31 | \( 1 + 9.30T + 31T^{2} \) |
| 41 | \( 1 + 2.50T + 41T^{2} \) |
| 43 | \( 1 - 2.92iT - 43T^{2} \) |
| 47 | \( 1 - 7.18iT - 47T^{2} \) |
| 53 | \( 1 + 13.8iT - 53T^{2} \) |
| 59 | \( 1 + 3.18T + 59T^{2} \) |
| 61 | \( 1 - 8.78T + 61T^{2} \) |
| 67 | \( 1 + 15.2iT - 67T^{2} \) |
| 71 | \( 1 + 6.65T + 71T^{2} \) |
| 73 | \( 1 - 11.9iT - 73T^{2} \) |
| 79 | \( 1 - 0.797T + 79T^{2} \) |
| 83 | \( 1 + 7.72iT - 83T^{2} \) |
| 89 | \( 1 + 14.8T + 89T^{2} \) |
| 97 | \( 1 - 13.0iT - 97T^{2} \) |
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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
−9.490711548029505911507434267565, −8.608997374787592777465131984633, −7.72386575922939745142261876458, −6.99595446092550162457999984939, −6.01221144591527665361670341924, −5.08201040293757682389345104764, −3.77567590844139524435815484987, −2.82313477782778467537032769892, −2.04603127975099496166431020128, −0.18010721482245116791057655205,
1.66217046851468945730540157287, 3.72586691924351620731351173842, 4.12532904581334507702006683346, 5.34314020679153977273731953578, 5.76147505747539516801835042089, 7.17580412668296364084963805995, 7.74768781357659008225257939690, 8.791374236878671233133090701168, 9.199665417964938960289680415430, 10.10672850703432983000067091951