| L(s) = 1 | + 2.20·2-s + 1.14·3-s + 2.87·4-s − 0.436·5-s + 2.53·6-s + 3.86·7-s + 1.94·8-s − 1.68·9-s − 0.964·10-s − 0.587·11-s + 3.30·12-s + 13-s + 8.54·14-s − 0.501·15-s − 1.46·16-s + 6.88·17-s − 3.71·18-s − 0.710·19-s − 1.25·20-s + 4.43·21-s − 1.29·22-s + 0.0670·23-s + 2.22·24-s − 4.80·25-s + 2.20·26-s − 5.37·27-s + 11.1·28-s + ⋯ |
| L(s) = 1 | + 1.56·2-s + 0.662·3-s + 1.43·4-s − 0.195·5-s + 1.03·6-s + 1.46·7-s + 0.686·8-s − 0.560·9-s − 0.305·10-s − 0.177·11-s + 0.954·12-s + 0.277·13-s + 2.28·14-s − 0.129·15-s − 0.366·16-s + 1.66·17-s − 0.876·18-s − 0.163·19-s − 0.281·20-s + 0.968·21-s − 0.276·22-s + 0.0139·23-s + 0.455·24-s − 0.961·25-s + 0.433·26-s − 1.03·27-s + 2.10·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.842372866\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.842372866\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 - T \) |
| 79 | \( 1 + T \) |
| good | 2 | \( 1 - 2.20T + 2T^{2} \) |
| 3 | \( 1 - 1.14T + 3T^{2} \) |
| 5 | \( 1 + 0.436T + 5T^{2} \) |
| 7 | \( 1 - 3.86T + 7T^{2} \) |
| 11 | \( 1 + 0.587T + 11T^{2} \) |
| 17 | \( 1 - 6.88T + 17T^{2} \) |
| 19 | \( 1 + 0.710T + 19T^{2} \) |
| 23 | \( 1 - 0.0670T + 23T^{2} \) |
| 29 | \( 1 - 3.45T + 29T^{2} \) |
| 31 | \( 1 - 3.61T + 31T^{2} \) |
| 37 | \( 1 + 3.99T + 37T^{2} \) |
| 41 | \( 1 + 4.14T + 41T^{2} \) |
| 43 | \( 1 + 1.98T + 43T^{2} \) |
| 47 | \( 1 + 0.00586T + 47T^{2} \) |
| 53 | \( 1 + 2.42T + 53T^{2} \) |
| 59 | \( 1 + 2.78T + 59T^{2} \) |
| 61 | \( 1 + 9.08T + 61T^{2} \) |
| 67 | \( 1 - 3.12T + 67T^{2} \) |
| 71 | \( 1 + 15.7T + 71T^{2} \) |
| 73 | \( 1 - 1.79T + 73T^{2} \) |
| 83 | \( 1 - 15.6T + 83T^{2} \) |
| 89 | \( 1 + 4.31T + 89T^{2} \) |
| 97 | \( 1 + 0.421T + 97T^{2} \) |
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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
−10.13265336038672893156009017024, −8.878177889301624465704943643281, −8.089039936033716768900720263677, −7.52727471807663136618229468370, −6.18918193254127089359039447676, −5.40528330959348553528513621063, −4.70011867777355774652205360288, −3.69420891174305979739588437880, −2.90740740049302643212574112496, −1.73682712041574853977088996838,
1.73682712041574853977088996838, 2.90740740049302643212574112496, 3.69420891174305979739588437880, 4.70011867777355774652205360288, 5.40528330959348553528513621063, 6.18918193254127089359039447676, 7.52727471807663136618229468370, 8.089039936033716768900720263677, 8.878177889301624465704943643281, 10.13265336038672893156009017024
