L(s) = 1 | + 3-s − 1.41·7-s + 9-s + 1.41·13-s − 1.41·21-s + 25-s + 27-s + 1.41·31-s − 1.41·37-s + 1.41·39-s + 1.00·49-s + 1.41·61-s − 1.41·63-s − 2·67-s + 75-s + 1.41·79-s + 81-s − 2.00·91-s + 1.41·93-s + 1.41·103-s − 1.41·109-s − 1.41·111-s + 1.41·117-s + ⋯ |
L(s) = 1 | + 3-s − 1.41·7-s + 9-s + 1.41·13-s − 1.41·21-s + 25-s + 27-s + 1.41·31-s − 1.41·37-s + 1.41·39-s + 1.00·49-s + 1.41·61-s − 1.41·63-s − 2·67-s + 75-s + 1.41·79-s + 81-s − 2.00·91-s + 1.41·93-s + 1.41·103-s − 1.41·109-s − 1.41·111-s + 1.41·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.686269070\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.686269070\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
good | 5 | \( 1 - T^{2} \) |
| 7 | \( 1 + 1.41T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - 1.41T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - 1.41T + T^{2} \) |
| 37 | \( 1 + 1.41T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - 1.41T + T^{2} \) |
| 67 | \( 1 + 2T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 - 1.41T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−8.794205864114964824314411705256, −8.399665349446361556498060224534, −7.37499643241552464779916070214, −6.62116947691019559362048009594, −6.14473742322609602884630383998, −4.92424057702025312292699280819, −3.85197022602547665222686457463, −3.33011194095108258170088571177, −2.54625549014459716940512483532, −1.20977330238521163189448484916,
1.20977330238521163189448484916, 2.54625549014459716940512483532, 3.33011194095108258170088571177, 3.85197022602547665222686457463, 4.92424057702025312292699280819, 6.14473742322609602884630383998, 6.62116947691019559362048009594, 7.37499643241552464779916070214, 8.399665349446361556498060224534, 8.794205864114964824314411705256