L(s) = 1 | + 1.61·2-s + 1.23·3-s + 0.618·4-s + 2.00·6-s + 7-s − 2.23·8-s − 1.47·9-s + 4.23·11-s + 0.763·12-s − 3.23·13-s + 1.61·14-s − 4.85·16-s − 6.47·17-s − 2.38·18-s + 4.47·19-s + 1.23·21-s + 6.85·22-s + 1.76·23-s − 2.76·24-s − 5.23·26-s − 5.52·27-s + 0.618·28-s + 5·29-s − 9.70·31-s − 3.38·32-s + 5.23·33-s − 10.4·34-s + ⋯ |
L(s) = 1 | + 1.14·2-s + 0.713·3-s + 0.309·4-s + 0.816·6-s + 0.377·7-s − 0.790·8-s − 0.490·9-s + 1.27·11-s + 0.220·12-s − 0.897·13-s + 0.432·14-s − 1.21·16-s − 1.56·17-s − 0.561·18-s + 1.02·19-s + 0.269·21-s + 1.46·22-s + 0.367·23-s − 0.564·24-s − 1.02·26-s − 1.06·27-s + 0.116·28-s + 0.928·29-s − 1.74·31-s − 0.597·32-s + 0.911·33-s − 1.79·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.149166497\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.149166497\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 2 | \( 1 - 1.61T + 2T^{2} \) |
| 3 | \( 1 - 1.23T + 3T^{2} \) |
| 11 | \( 1 - 4.23T + 11T^{2} \) |
| 13 | \( 1 + 3.23T + 13T^{2} \) |
| 17 | \( 1 + 6.47T + 17T^{2} \) |
| 19 | \( 1 - 4.47T + 19T^{2} \) |
| 23 | \( 1 - 1.76T + 23T^{2} \) |
| 29 | \( 1 - 5T + 29T^{2} \) |
| 31 | \( 1 + 9.70T + 31T^{2} \) |
| 37 | \( 1 - 3T + 37T^{2} \) |
| 41 | \( 1 - 9.23T + 41T^{2} \) |
| 43 | \( 1 - 6.23T + 43T^{2} \) |
| 47 | \( 1 + 2T + 47T^{2} \) |
| 53 | \( 1 + 0.472T + 53T^{2} \) |
| 59 | \( 1 + 1.70T + 59T^{2} \) |
| 61 | \( 1 - 3.70T + 61T^{2} \) |
| 67 | \( 1 - 0.236T + 67T^{2} \) |
| 71 | \( 1 + 4.70T + 71T^{2} \) |
| 73 | \( 1 + 13.2T + 73T^{2} \) |
| 79 | \( 1 - 11.1T + 79T^{2} \) |
| 83 | \( 1 - 5.70T + 83T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 - 0.763T + 97T^{2} \) |
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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
−12.91809291978232345592203386122, −11.89979932095052491832649750440, −11.16180241119314248410350986533, −9.328962372612781598247437787214, −8.897933472475577529092045113779, −7.40007573503542778784844487620, −6.14821362082979655163210573056, −4.88346031161601700344980742774, −3.82033636486049561268810955661, −2.51928008739702714388271852683,
2.51928008739702714388271852683, 3.82033636486049561268810955661, 4.88346031161601700344980742774, 6.14821362082979655163210573056, 7.40007573503542778784844487620, 8.897933472475577529092045113779, 9.328962372612781598247437787214, 11.16180241119314248410350986533, 11.89979932095052491832649750440, 12.91809291978232345592203386122