L(s) = 1 | − 2-s − 2.61·3-s + 4-s − 3.61·5-s + 2.61·6-s − 1.23·7-s − 8-s + 3.85·9-s + 3.61·10-s + 11-s − 2.61·12-s + 6.47·13-s + 1.23·14-s + 9.47·15-s + 16-s + 1.85·17-s − 3.85·18-s − 1.61·19-s − 3.61·20-s + 3.23·21-s − 22-s + 0.381·23-s + 2.61·24-s + 8.09·25-s − 6.47·26-s − 2.23·27-s − 1.23·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.51·3-s + 0.5·4-s − 1.61·5-s + 1.06·6-s − 0.467·7-s − 0.353·8-s + 1.28·9-s + 1.14·10-s + 0.301·11-s − 0.755·12-s + 1.79·13-s + 0.330·14-s + 2.44·15-s + 0.250·16-s + 0.449·17-s − 0.908·18-s − 0.371·19-s − 0.809·20-s + 0.706·21-s − 0.213·22-s + 0.0796·23-s + 0.534·24-s + 1.61·25-s − 1.26·26-s − 0.430·27-s − 0.233·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 946 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 946 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 43 | \( 1 - T \) |
good | 3 | \( 1 + 2.61T + 3T^{2} \) |
| 5 | \( 1 + 3.61T + 5T^{2} \) |
| 7 | \( 1 + 1.23T + 7T^{2} \) |
| 13 | \( 1 - 6.47T + 13T^{2} \) |
| 17 | \( 1 - 1.85T + 17T^{2} \) |
| 19 | \( 1 + 1.61T + 19T^{2} \) |
| 23 | \( 1 - 0.381T + 23T^{2} \) |
| 29 | \( 1 + 2.61T + 29T^{2} \) |
| 31 | \( 1 + 5.85T + 31T^{2} \) |
| 37 | \( 1 - 7.85T + 37T^{2} \) |
| 41 | \( 1 + 6.38T + 41T^{2} \) |
| 47 | \( 1 + 4.61T + 47T^{2} \) |
| 53 | \( 1 - 0.472T + 53T^{2} \) |
| 59 | \( 1 - 8.94T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 15.7T + 67T^{2} \) |
| 71 | \( 1 + 7.23T + 71T^{2} \) |
| 73 | \( 1 - 12.9T + 73T^{2} \) |
| 79 | \( 1 + 6.85T + 79T^{2} \) |
| 83 | \( 1 + 13.2T + 83T^{2} \) |
| 89 | \( 1 + 5.52T + 89T^{2} \) |
| 97 | \( 1 + 8.85T + 97T^{2} \) |
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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
−9.767353658324036430689031918809, −8.662744983895226540056583853105, −7.978177573011484266552907284584, −6.97618019232470752147996817615, −6.36075816285336722507318275446, −5.45944879803517644616149562694, −4.16554524742773002495049483789, −3.42221274849502947780405962817, −1.16144279182738854892177266246, 0,
1.16144279182738854892177266246, 3.42221274849502947780405962817, 4.16554524742773002495049483789, 5.45944879803517644616149562694, 6.36075816285336722507318275446, 6.97618019232470752147996817615, 7.978177573011484266552907284584, 8.662744983895226540056583853105, 9.767353658324036430689031918809