L(s) = 1 | + 0.367·2-s + 3-s − 1.86·4-s − 0.382·5-s + 0.367·6-s + 2.18·7-s − 1.42·8-s + 9-s − 0.140·10-s − 5.51·11-s − 1.86·12-s − 2.93·13-s + 0.801·14-s − 0.382·15-s + 3.20·16-s − 17-s + 0.367·18-s + 5.52·19-s + 0.713·20-s + 2.18·21-s − 2.02·22-s − 0.477·23-s − 1.42·24-s − 4.85·25-s − 1.07·26-s + 27-s − 4.06·28-s + ⋯ |
L(s) = 1 | + 0.259·2-s + 0.577·3-s − 0.932·4-s − 0.171·5-s + 0.150·6-s + 0.824·7-s − 0.502·8-s + 0.333·9-s − 0.0444·10-s − 1.66·11-s − 0.538·12-s − 0.812·13-s + 0.214·14-s − 0.0988·15-s + 0.801·16-s − 0.242·17-s + 0.0866·18-s + 1.26·19-s + 0.159·20-s + 0.475·21-s − 0.432·22-s − 0.0994·23-s − 0.289·24-s − 0.970·25-s − 0.211·26-s + 0.192·27-s − 0.768·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.779633117\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.779633117\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 79 | \( 1 + T \) |
good | 2 | \( 1 - 0.367T + 2T^{2} \) |
| 5 | \( 1 + 0.382T + 5T^{2} \) |
| 7 | \( 1 - 2.18T + 7T^{2} \) |
| 11 | \( 1 + 5.51T + 11T^{2} \) |
| 13 | \( 1 + 2.93T + 13T^{2} \) |
| 19 | \( 1 - 5.52T + 19T^{2} \) |
| 23 | \( 1 + 0.477T + 23T^{2} \) |
| 29 | \( 1 - 3.54T + 29T^{2} \) |
| 31 | \( 1 - 0.807T + 31T^{2} \) |
| 37 | \( 1 - 1.01T + 37T^{2} \) |
| 41 | \( 1 - 9.86T + 41T^{2} \) |
| 43 | \( 1 + 0.423T + 43T^{2} \) |
| 47 | \( 1 - 4.89T + 47T^{2} \) |
| 53 | \( 1 - 9.10T + 53T^{2} \) |
| 59 | \( 1 - 2.89T + 59T^{2} \) |
| 61 | \( 1 + 5.87T + 61T^{2} \) |
| 67 | \( 1 + 4.06T + 67T^{2} \) |
| 71 | \( 1 - 16.3T + 71T^{2} \) |
| 73 | \( 1 - 1.50T + 73T^{2} \) |
| 83 | \( 1 - 2.84T + 83T^{2} \) |
| 89 | \( 1 - 2.36T + 89T^{2} \) |
| 97 | \( 1 + 2.24T + 97T^{2} \) |
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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
−8.344138524287674653664184067017, −7.69834399543782339559959366370, −7.45321149639364741396059466081, −5.98733717896745943422784721641, −5.18148992365571054022799536421, −4.77962231406367144452236722520, −3.92599055361483401000504018712, −2.95457596825068275814005436714, −2.20212430313282143474824172754, −0.70682954268310407465592720024,
0.70682954268310407465592720024, 2.20212430313282143474824172754, 2.95457596825068275814005436714, 3.92599055361483401000504018712, 4.77962231406367144452236722520, 5.18148992365571054022799536421, 5.98733717896745943422784721641, 7.45321149639364741396059466081, 7.69834399543782339559959366370, 8.344138524287674653664184067017