L(s) = 1 | − 2-s − 3.08·3-s + 4-s + 5-s + 3.08·6-s + 4.56·7-s − 8-s + 6.53·9-s − 10-s − 3.21·11-s − 3.08·12-s − 13-s − 4.56·14-s − 3.08·15-s + 16-s − 4.97·17-s − 6.53·18-s − 7.48·19-s + 20-s − 14.0·21-s + 3.21·22-s + 8.03·23-s + 3.08·24-s + 25-s + 26-s − 10.9·27-s + 4.56·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.78·3-s + 0.5·4-s + 0.447·5-s + 1.26·6-s + 1.72·7-s − 0.353·8-s + 2.17·9-s − 0.316·10-s − 0.969·11-s − 0.891·12-s − 0.277·13-s − 1.21·14-s − 0.797·15-s + 0.250·16-s − 1.20·17-s − 1.53·18-s − 1.71·19-s + 0.223·20-s − 3.07·21-s + 0.685·22-s + 1.67·23-s + 0.630·24-s + 0.200·25-s + 0.196·26-s − 2.09·27-s + 0.862·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7462723229\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7462723229\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 + T \) |
good | 3 | \( 1 + 3.08T + 3T^{2} \) |
| 7 | \( 1 - 4.56T + 7T^{2} \) |
| 11 | \( 1 + 3.21T + 11T^{2} \) |
| 17 | \( 1 + 4.97T + 17T^{2} \) |
| 19 | \( 1 + 7.48T + 19T^{2} \) |
| 23 | \( 1 - 8.03T + 23T^{2} \) |
| 29 | \( 1 - 4.31T + 29T^{2} \) |
| 37 | \( 1 + 5.02T + 37T^{2} \) |
| 41 | \( 1 - 7.36T + 41T^{2} \) |
| 43 | \( 1 + 1.97T + 43T^{2} \) |
| 47 | \( 1 - 3.07T + 47T^{2} \) |
| 53 | \( 1 - 3.25T + 53T^{2} \) |
| 59 | \( 1 + 4.07T + 59T^{2} \) |
| 61 | \( 1 + 4.76T + 61T^{2} \) |
| 67 | \( 1 - 12.0T + 67T^{2} \) |
| 71 | \( 1 + 8.72T + 71T^{2} \) |
| 73 | \( 1 - 5.98T + 73T^{2} \) |
| 79 | \( 1 + 15.0T + 79T^{2} \) |
| 83 | \( 1 - 11.1T + 83T^{2} \) |
| 89 | \( 1 - 2.41T + 89T^{2} \) |
| 97 | \( 1 - 13.0T + 97T^{2} \) |
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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
−8.500866480992754523230203895195, −7.59757470628698769789180576482, −6.93652833375662203873970825091, −6.27720117833004633509122420564, −5.43492755299602695899024468659, −4.85068127066495617568304338342, −4.38367364662254327354058075288, −2.44851908760400592195014992417, −1.68453761440257314613689645465, −0.61472584774665224096801135227,
0.61472584774665224096801135227, 1.68453761440257314613689645465, 2.44851908760400592195014992417, 4.38367364662254327354058075288, 4.85068127066495617568304338342, 5.43492755299602695899024468659, 6.27720117833004633509122420564, 6.93652833375662203873970825091, 7.59757470628698769789180576482, 8.500866480992754523230203895195