L(s) = 1 | + (0.923 − 0.382i)3-s + (0.541 + 0.541i)7-s + (0.707 − 0.707i)9-s + (0.707 + 1.70i)13-s + (0.541 + 1.30i)19-s + (0.707 + 0.292i)21-s + (−0.707 − 0.707i)25-s + (0.382 − 0.923i)27-s − 1.84·31-s + (−0.292 + 0.707i)37-s + (1.30 + 1.30i)39-s + (−1.30 − 0.541i)43-s − 0.414i·49-s + (1 + 0.999i)57-s + (1.70 − 0.707i)61-s + ⋯ |
L(s) = 1 | + (0.923 − 0.382i)3-s + (0.541 + 0.541i)7-s + (0.707 − 0.707i)9-s + (0.707 + 1.70i)13-s + (0.541 + 1.30i)19-s + (0.707 + 0.292i)21-s + (−0.707 − 0.707i)25-s + (0.382 − 0.923i)27-s − 1.84·31-s + (−0.292 + 0.707i)37-s + (1.30 + 1.30i)39-s + (−1.30 − 0.541i)43-s − 0.414i·49-s + (1 + 0.999i)57-s + (1.70 − 0.707i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.890394671\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.890394671\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.923 + 0.382i)T \) |
good | 5 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 7 | \( 1 + (-0.541 - 0.541i)T + iT^{2} \) |
| 11 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 13 | \( 1 + (-0.707 - 1.70i)T + (-0.707 + 0.707i)T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + (-0.541 - 1.30i)T + (-0.707 + 0.707i)T^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 31 | \( 1 + 1.84T + T^{2} \) |
| 37 | \( 1 + (0.292 - 0.707i)T + (-0.707 - 0.707i)T^{2} \) |
| 41 | \( 1 + iT^{2} \) |
| 43 | \( 1 + (1.30 + 0.541i)T + (0.707 + 0.707i)T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 59 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 61 | \( 1 + (-1.70 + 0.707i)T + (0.707 - 0.707i)T^{2} \) |
| 67 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 71 | \( 1 - iT^{2} \) |
| 73 | \( 1 + (-1 + i)T - iT^{2} \) |
| 79 | \( 1 + 1.84iT - T^{2} \) |
| 83 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 89 | \( 1 - iT^{2} \) |
| 97 | \( 1 - 1.41T + T^{2} \) |
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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
−8.749648000577475875809439105592, −8.310751357723339832493307340480, −7.51644688012892835444046254650, −6.74036007584860157193711349534, −6.02463057018790341164559707738, −5.02369240462249214508254048316, −3.96450667028965365726890161802, −3.44566655506188047664935395100, −2.00643615845684064520767421890, −1.69060692272440831890103879441,
1.19240539893880441963275024960, 2.41021240239924764305888627731, 3.38978873110789633835030681482, 3.92412439430049151967456876559, 5.07979410923344842253636417614, 5.52611477256884100719604481294, 6.88939503237393684201211868127, 7.55901442687906423497334744974, 8.100655426338341243832379393291, 8.821348161481631576079682579304