L(s) = 1 | + (0.707 − 0.707i)3-s + (−1.41 − 1.41i)7-s − 1.00i·9-s − 2.00·21-s + (−0.707 − 0.707i)27-s + (1.41 − 1.41i)43-s + 3.00i·49-s + 2·61-s + (−1.41 + 1.41i)63-s + (1.41 + 1.41i)67-s − 1.00·81-s + (−1.41 + 1.41i)103-s − 2i·109-s + ⋯ |
L(s) = 1 | + (0.707 − 0.707i)3-s + (−1.41 − 1.41i)7-s − 1.00i·9-s − 2.00·21-s + (−0.707 − 0.707i)27-s + (1.41 − 1.41i)43-s + 3.00i·49-s + 2·61-s + (−1.41 + 1.41i)63-s + (1.41 + 1.41i)67-s − 1.00·81-s + (−1.41 + 1.41i)103-s − 2i·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.072760838\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.072760838\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (1.41 + 1.41i)T + iT^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + (-1.41 + 1.41i)T - iT^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - 2T + T^{2} \) |
| 67 | \( 1 + (-1.41 - 1.41i)T + iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 - iT^{2} \) |
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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
−9.707755913366339168086885107773, −8.930370203106244238723660490045, −7.959703963676752963041471858506, −7.09667085380094546767123062037, −6.74884801600807357590526138135, −5.72985770508222206291467081173, −4.12131701679073215332867686006, −3.52784041319034434727517903042, −2.47595419092273291458371071096, −0.874279031667763229495285711211,
2.28492571135972876086159921573, 3.01609200972239756683414937444, 3.90709732369647258909861137397, 5.12146938351550802448656829725, 5.92011309673505568806087375931, 6.80058199737320817471580939049, 7.981540232315510327685759725409, 8.741782914603429348453313735309, 9.452254246984479316148879186376, 9.812557536948061461022463574561