L(s) = 1 | − 23·7-s + 191·13-s − 647·19-s + 625·25-s − 194·31-s + 2.59e3·37-s + 3.21e3·43-s − 1.87e3·49-s − 5.23e3·61-s + 8.80e3·67-s + 9.79e3·73-s + 1.23e4·79-s − 4.39e3·91-s + 9.74e3·97-s − 3.43e3·103-s + 2.20e4·109-s + ⋯ |
L(s) = 1 | − 0.469·7-s + 1.13·13-s − 1.79·19-s + 25-s − 0.201·31-s + 1.89·37-s + 1.73·43-s − 0.779·49-s − 1.40·61-s + 1.96·67-s + 1.83·73-s + 1.98·79-s − 0.530·91-s + 1.03·97-s − 0.323·103-s + 1.85·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(1.864361448\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.864361448\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 7 | \( 1 + 23 T + p^{4} T^{2} \) |
| 11 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 13 | \( 1 - 191 T + p^{4} T^{2} \) |
| 17 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 19 | \( 1 + 647 T + p^{4} T^{2} \) |
| 23 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 29 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 31 | \( 1 + 194 T + p^{4} T^{2} \) |
| 37 | \( 1 - 2591 T + p^{4} T^{2} \) |
| 41 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 43 | \( 1 - 3214 T + p^{4} T^{2} \) |
| 47 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 53 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 59 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 61 | \( 1 + 5233 T + p^{4} T^{2} \) |
| 67 | \( 1 - 8809 T + p^{4} T^{2} \) |
| 71 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 73 | \( 1 - 9791 T + p^{4} T^{2} \) |
| 79 | \( 1 - 12361 T + p^{4} T^{2} \) |
| 83 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 89 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 97 | \( 1 - 9743 T + p^{4} T^{2} \) |
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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
−10.74266658094348185978317507486, −9.562741325019853304485816935342, −8.747627603800083143161519127330, −7.88280616160315803340710397783, −6.59447469445795556269777830308, −6.02338393396236202075333404311, −4.60950460776271367794115958120, −3.60108894615956467599697521594, −2.30957459986660492721802569466, −0.78151994581548855163194047427,
0.78151994581548855163194047427, 2.30957459986660492721802569466, 3.60108894615956467599697521594, 4.60950460776271367794115958120, 6.02338393396236202075333404311, 6.59447469445795556269777830308, 7.88280616160315803340710397783, 8.747627603800083143161519127330, 9.562741325019853304485816935342, 10.74266658094348185978317507486