L(s) = 1 | + (−0.270 + 0.270i)5-s + 7-s + (3.03 + 3.03i)11-s + (1.28 − 1.28i)13-s + 5.15i·17-s + (5.55 + 5.55i)19-s − 5.69i·23-s + 4.85i·25-s + (−1.94 − 1.94i)29-s + 0.936i·31-s + (−0.270 + 0.270i)35-s + (−1.52 − 1.52i)37-s − 12.4·41-s + (0.346 − 0.346i)43-s + 9.71·47-s + ⋯ |
L(s) = 1 | + (−0.120 + 0.120i)5-s + 0.377·7-s + (0.915 + 0.915i)11-s + (0.357 − 0.357i)13-s + 1.25i·17-s + (1.27 + 1.27i)19-s − 1.18i·23-s + 0.970i·25-s + (−0.360 − 0.360i)29-s + 0.168i·31-s + (−0.0456 + 0.0456i)35-s + (−0.251 − 0.251i)37-s − 1.94·41-s + (0.0528 − 0.0528i)43-s + 1.41·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.272 - 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.272 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.981182327\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.981182327\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
good | 5 | \( 1 + (0.270 - 0.270i)T - 5iT^{2} \) |
| 11 | \( 1 + (-3.03 - 3.03i)T + 11iT^{2} \) |
| 13 | \( 1 + (-1.28 + 1.28i)T - 13iT^{2} \) |
| 17 | \( 1 - 5.15iT - 17T^{2} \) |
| 19 | \( 1 + (-5.55 - 5.55i)T + 19iT^{2} \) |
| 23 | \( 1 + 5.69iT - 23T^{2} \) |
| 29 | \( 1 + (1.94 + 1.94i)T + 29iT^{2} \) |
| 31 | \( 1 - 0.936iT - 31T^{2} \) |
| 37 | \( 1 + (1.52 + 1.52i)T + 37iT^{2} \) |
| 41 | \( 1 + 12.4T + 41T^{2} \) |
| 43 | \( 1 + (-0.346 + 0.346i)T - 43iT^{2} \) |
| 47 | \( 1 - 9.71T + 47T^{2} \) |
| 53 | \( 1 + (7.69 - 7.69i)T - 53iT^{2} \) |
| 59 | \( 1 + (2.03 + 2.03i)T + 59iT^{2} \) |
| 61 | \( 1 + (-5.86 + 5.86i)T - 61iT^{2} \) |
| 67 | \( 1 + (10.7 + 10.7i)T + 67iT^{2} \) |
| 71 | \( 1 - 12.5iT - 71T^{2} \) |
| 73 | \( 1 + 7.37iT - 73T^{2} \) |
| 79 | \( 1 - 13.0iT - 79T^{2} \) |
| 83 | \( 1 + (2.53 - 2.53i)T - 83iT^{2} \) |
| 89 | \( 1 - 7.04T + 89T^{2} \) |
| 97 | \( 1 - 0.334T + 97T^{2} \) |
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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
−8.537531965940589568795086623212, −7.84966356763933698218125573041, −7.20339358307379128084519610976, −6.37120821134807010645679621499, −5.68500070944280912848225661112, −4.81155825414483646139517159117, −3.91754341462580738808879301300, −3.35478419998130131828402605722, −1.97059683817187939592787305628, −1.26158824788382511037710429344,
0.62208671191150394248846859435, 1.60371677166777535970200763055, 2.91839208075649837540440234225, 3.57064459723645454069456204432, 4.58819619183169371368258657057, 5.25827168566576871028606841263, 6.05258042126165857313568737722, 6.97863736381211172249608322703, 7.41619889681047625102390525532, 8.433249848936440340324062966815