L(s) = 1 | + (0.707 − 0.707i)2-s − 1.00i·4-s + (0.707 + 0.707i)7-s + (−0.707 − 0.707i)8-s − 4.82i·11-s + (−3.41 + 3.41i)13-s + 1.00·14-s − 1.00·16-s + (1.58 − 1.58i)17-s − 7.07i·19-s + (−3.41 − 3.41i)22-s + (4.82 + 4.82i)23-s + 4.82i·26-s + (0.707 − 0.707i)28-s − 3.17·29-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s − 0.500i·4-s + (0.267 + 0.267i)7-s + (−0.250 − 0.250i)8-s − 1.45i·11-s + (−0.946 + 0.946i)13-s + 0.267·14-s − 0.250·16-s + (0.384 − 0.384i)17-s − 1.62i·19-s + (−0.727 − 0.727i)22-s + (1.00 + 1.00i)23-s + 0.946i·26-s + (0.133 − 0.133i)28-s − 0.588·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.920 + 0.391i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.920 + 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.532248963\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.532248963\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.707 - 0.707i)T \) |
good | 11 | \( 1 + 4.82iT - 11T^{2} \) |
| 13 | \( 1 + (3.41 - 3.41i)T - 13iT^{2} \) |
| 17 | \( 1 + (-1.58 + 1.58i)T - 17iT^{2} \) |
| 19 | \( 1 + 7.07iT - 19T^{2} \) |
| 23 | \( 1 + (-4.82 - 4.82i)T + 23iT^{2} \) |
| 29 | \( 1 + 3.17T + 29T^{2} \) |
| 31 | \( 1 + 8.82T + 31T^{2} \) |
| 37 | \( 1 + (6.07 + 6.07i)T + 37iT^{2} \) |
| 41 | \( 1 - 4.82iT - 41T^{2} \) |
| 43 | \( 1 + (-6.41 + 6.41i)T - 43iT^{2} \) |
| 47 | \( 1 + (-6.41 + 6.41i)T - 47iT^{2} \) |
| 53 | \( 1 + (4.82 + 4.82i)T + 53iT^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 - 6.58T + 61T^{2} \) |
| 67 | \( 1 + (-0.414 - 0.414i)T + 67iT^{2} \) |
| 71 | \( 1 - 12.2iT - 71T^{2} \) |
| 73 | \( 1 + (6 - 6i)T - 73iT^{2} \) |
| 79 | \( 1 + 13.3iT - 79T^{2} \) |
| 83 | \( 1 + (9.65 + 9.65i)T + 83iT^{2} \) |
| 89 | \( 1 + 13.3T + 89T^{2} \) |
| 97 | \( 1 + (8.24 + 8.24i)T + 97iT^{2} \) |
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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
−8.697538581212803081723856836819, −7.25925375112465015905152001673, −7.07219654368185706993912186260, −5.69457033357114548571134118197, −5.37446865885741896620333997673, −4.44772554609412277626013988481, −3.48593459473262998273581175468, −2.73638249043570679269708135855, −1.73929204026715908590831042123, −0.37819456128522002662235569882,
1.53175132943956846750161837661, 2.61622580916779869568255226057, 3.65284204727890436855739749492, 4.48001283313385336046514333623, 5.18075396463620687253856508172, 5.86788169274077302047597449761, 6.86233358219204620456258069172, 7.57089556821765262214746383896, 7.87234798792932557341452187373, 8.936437740353928077996887872229