| L(s) = 1 | + i·2-s + 3-s − 4-s + 3i·5-s + i·6-s − i·8-s − 2·9-s − 3·10-s − 12-s + (−2 − 3i)13-s + 3i·15-s + 16-s − 3·17-s − 2i·18-s + 6i·19-s − 3i·20-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.577·3-s − 0.5·4-s + 1.34i·5-s + 0.408i·6-s − 0.353i·8-s − 0.666·9-s − 0.948·10-s − 0.288·12-s + (−0.554 − 0.832i)13-s + 0.774i·15-s + 0.250·16-s − 0.727·17-s − 0.471i·18-s + 1.37i·19-s − 0.670i·20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7353881042\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7353881042\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 7 | \( 1 \) |
| 13 | \( 1 + (2 + 3i)T \) |
| good | 3 | \( 1 - T + 3T^{2} \) |
| 5 | \( 1 - 3iT - 5T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 - 6iT - 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 3iT - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + T + 43T^{2} \) |
| 47 | \( 1 + 3iT - 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 6iT - 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 + 12iT - 67T^{2} \) |
| 71 | \( 1 - 15iT - 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 - 6iT - 89T^{2} \) |
| 97 | \( 1 - 12iT - 97T^{2} \) |
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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
−10.10258568979940088599983144930, −9.303174547707715162015696818145, −8.146136132144137611464316936936, −7.903981653463609263376344440170, −6.86749199959062239285900991994, −6.14828936113467308883189587520, −5.36118951509767446478862873306, −3.97375178412471925947689655489, −3.15240974857690971116427223197, −2.21328283889656046556872531294,
0.26443151766956092446487921182, 1.82401273910339278189209075082, 2.69900066446135815829022500786, 4.01188494039394235887480642625, 4.69998363903456048616341944583, 5.54200561231411218410755730686, 6.75484292360192518643024258134, 7.965012769356168601218124801149, 8.596184817503442915513823967274, 9.247505587401910499150306161007
