L(s) = 1 | + (0.965 + 0.258i)5-s + (−0.866 + 0.5i)7-s + i·13-s + (0.707 − 1.22i)17-s + (0.5 + 0.866i)19-s + (0.707 + 1.22i)23-s + (0.866 + 0.499i)25-s − 1.41i·29-s + (−0.5 + 0.866i)31-s + (−0.965 + 0.258i)35-s + (−0.866 + 0.5i)37-s − i·43-s + (−0.707 − 1.22i)47-s + (0.499 − 0.866i)49-s + (1.22 + 0.707i)59-s + ⋯ |
L(s) = 1 | + (0.965 + 0.258i)5-s + (−0.866 + 0.5i)7-s + i·13-s + (0.707 − 1.22i)17-s + (0.5 + 0.866i)19-s + (0.707 + 1.22i)23-s + (0.866 + 0.499i)25-s − 1.41i·29-s + (−0.5 + 0.866i)31-s + (−0.965 + 0.258i)35-s + (−0.866 + 0.5i)37-s − i·43-s + (−0.707 − 1.22i)47-s + (0.499 − 0.866i)49-s + (1.22 + 0.707i)59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.807 - 0.589i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.807 - 0.589i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.168101676\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.168101676\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.965 - 0.258i)T \) |
| 7 | \( 1 + (0.866 - 0.5i)T \) |
good | 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 - iT - T^{2} \) |
| 17 | \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + 1.41iT - T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + iT - T^{2} \) |
| 47 | \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.681595137072751959977228691293, −9.477641216550112978684598829079, −8.542924204115012052336085988043, −7.24554248827108205254862584517, −6.73650030631645077135405533624, −5.69215674305709532052304427878, −5.20372779862400958204120747068, −3.67992659247001805644923405695, −2.81411162461294760922285626071, −1.66226024200484674677880923368,
1.15012761045758731498014932025, 2.67052586483773711914503354483, 3.53826080062410070942245004976, 4.82374073316734900692096414808, 5.67535780956904547398390404204, 6.43443574509573157210076308860, 7.23654912812274273790341480164, 8.287058446359303676011524265594, 9.103839600177126566752024453716, 9.845916634135144590772693272075