L(s) = 1 | − 2·5-s − 11-s − 6·13-s − 3·17-s − 5·19-s + 23-s − 25-s − 5·29-s + 8·31-s + 7·37-s − 6·41-s + 11·43-s + 9·47-s + 2·53-s + 2·55-s + 3·59-s + 4·61-s + 12·65-s + 6·67-s − 71-s + 14·73-s + 4·79-s + 16·83-s + 6·85-s + 8·89-s + 10·95-s + 11·97-s + ⋯ |
L(s) = 1 | − 0.894·5-s − 0.301·11-s − 1.66·13-s − 0.727·17-s − 1.14·19-s + 0.208·23-s − 1/5·25-s − 0.928·29-s + 1.43·31-s + 1.15·37-s − 0.937·41-s + 1.67·43-s + 1.31·47-s + 0.274·53-s + 0.269·55-s + 0.390·59-s + 0.512·61-s + 1.48·65-s + 0.733·67-s − 0.118·71-s + 1.63·73-s + 0.450·79-s + 1.75·83-s + 0.650·85-s + 0.847·89-s + 1.02·95-s + 1.11·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.414482738\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.414482738\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 - 11 T + p T^{2} \) |
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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
−12.43850065522438, −12.25608079768745, −11.85985901248702, −11.19621664954989, −10.95979336409753, −10.42229782605783, −9.884427508376182, −9.482663078627285, −8.978951528438580, −8.422370556483569, −7.996287450661295, −7.501074945674271, −7.260050748596858, −6.571349007914367, −6.199571544042005, −5.521080986052096, −4.875318803414061, −4.623629855309855, −3.991164545129751, −3.682822633936590, −2.774622407744285, −2.316391220408311, −2.059963246021738, −0.8038015755879678, −0.4119441958754088,
0.4119441958754088, 0.8038015755879678, 2.059963246021738, 2.316391220408311, 2.774622407744285, 3.682822633936590, 3.991164545129751, 4.623629855309855, 4.875318803414061, 5.521080986052096, 6.199571544042005, 6.571349007914367, 7.260050748596858, 7.501074945674271, 7.996287450661295, 8.422370556483569, 8.978951528438580, 9.482663078627285, 9.884427508376182, 10.42229782605783, 10.95979336409753, 11.19621664954989, 11.85985901248702, 12.25608079768745, 12.43850065522438