L(s) = 1 | + 2i·2-s + 5.04i·3-s − 4·4-s − 10.0·6-s + 5.03i·7-s − 8i·8-s + 1.58·9-s − 5.58·11-s − 20.1i·12-s − 62.7i·13-s − 10.0·14-s + 16·16-s − 19.7i·17-s + 3.17i·18-s − 158.·19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + 0.970i·3-s − 0.5·4-s − 0.685·6-s + 0.271i·7-s − 0.353i·8-s + 0.0588·9-s − 0.153·11-s − 0.485i·12-s − 1.33i·13-s − 0.192·14-s + 0.250·16-s − 0.281i·17-s + 0.0416i·18-s − 1.91·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.745360913\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.745360913\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 2iT \) |
| 5 | \( 1 \) |
| 23 | \( 1 - 23iT \) |
good | 3 | \( 1 - 5.04iT - 27T^{2} \) |
| 7 | \( 1 - 5.03iT - 343T^{2} \) |
| 11 | \( 1 + 5.58T + 1.33e3T^{2} \) |
| 13 | \( 1 + 62.7iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 19.7iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 158.T + 6.85e3T^{2} \) |
| 29 | \( 1 - 35.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 282.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 139. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 227.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 436. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 90.2iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 330. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 796.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 568.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 85.1iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 369.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 310. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 1.32e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 158. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 1.23e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 106. iT - 9.12e5T^{2} \) |
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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
−9.517392103449158155262467845472, −8.693768301426937709958809224597, −8.057104124941433409641839558454, −7.06184536201642773396412556968, −6.09564101553183982839185087836, −5.28703036751915787035029507582, −4.48511405111777581556994312739, −3.66307200059806434440105768441, −2.43349055533756103538051719865, −0.57611388679081061203647963712,
0.819325275377825849433317100462, 1.82830380867609371353478329579, 2.60689671665312358013737664324, 4.11648213665812794801499454667, 4.61058316557143288444632561395, 6.24841878112926095282061470450, 6.63131016244228753776142352463, 7.73082384823926420360272657662, 8.445157629495988417023764699533, 9.290305176328450450562048531623