L(s) = 1 | + (−1.63 + 1.08i)3-s + (1.08 − 2.63i)9-s + (0.617 − 0.923i)11-s + (0.382 + 0.923i)17-s + (1.30 − 0.541i)19-s + (−0.923 − 0.382i)25-s + (0.707 + 3.55i)27-s + 2.17i·33-s + (0.324 + 1.63i)41-s + (0.707 + 0.292i)43-s + (0.923 − 0.382i)49-s + (−1.63 − 1.08i)51-s + (−1.54 + 2.30i)57-s + (−0.292 + 0.707i)59-s + 1.84·67-s + ⋯ |
L(s) = 1 | + (−1.63 + 1.08i)3-s + (1.08 − 2.63i)9-s + (0.617 − 0.923i)11-s + (0.382 + 0.923i)17-s + (1.30 − 0.541i)19-s + (−0.923 − 0.382i)25-s + (0.707 + 3.55i)27-s + 2.17i·33-s + (0.324 + 1.63i)41-s + (0.707 + 0.292i)43-s + (0.923 − 0.382i)49-s + (−1.63 − 1.08i)51-s + (−1.54 + 2.30i)57-s + (−0.292 + 0.707i)59-s + 1.84·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 - 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 - 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6459040976\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6459040976\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 17 | \( 1 + (-0.382 - 0.923i)T \) |
good | 3 | \( 1 + (1.63 - 1.08i)T + (0.382 - 0.923i)T^{2} \) |
| 5 | \( 1 + (0.923 + 0.382i)T^{2} \) |
| 7 | \( 1 + (-0.923 + 0.382i)T^{2} \) |
| 11 | \( 1 + (-0.617 + 0.923i)T + (-0.382 - 0.923i)T^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 19 | \( 1 + (-1.30 + 0.541i)T + (0.707 - 0.707i)T^{2} \) |
| 23 | \( 1 + (-0.382 - 0.923i)T^{2} \) |
| 29 | \( 1 + (-0.923 - 0.382i)T^{2} \) |
| 31 | \( 1 + (0.382 - 0.923i)T^{2} \) |
| 37 | \( 1 + (0.382 - 0.923i)T^{2} \) |
| 41 | \( 1 + (-0.324 - 1.63i)T + (-0.923 + 0.382i)T^{2} \) |
| 43 | \( 1 + (-0.707 - 0.292i)T + (0.707 + 0.707i)T^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 59 | \( 1 + (0.292 - 0.707i)T + (-0.707 - 0.707i)T^{2} \) |
| 61 | \( 1 + (-0.923 + 0.382i)T^{2} \) |
| 67 | \( 1 - 1.84T + T^{2} \) |
| 71 | \( 1 + (-0.382 + 0.923i)T^{2} \) |
| 73 | \( 1 + (-0.216 + 1.08i)T + (-0.923 - 0.382i)T^{2} \) |
| 79 | \( 1 + (0.382 + 0.923i)T^{2} \) |
| 83 | \( 1 + (0.707 + 1.70i)T + (-0.707 + 0.707i)T^{2} \) |
| 89 | \( 1 + (-0.541 + 0.541i)T - iT^{2} \) |
| 97 | \( 1 + (-0.382 - 0.0761i)T + (0.923 + 0.382i)T^{2} \) |
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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
−10.19320500581123469234635598469, −9.601479319343728385396033285647, −8.793789814666751943810416868021, −7.51963736079871338446004782547, −6.34198389205981413176900828905, −5.92098223155904250441497932363, −5.06917553444302039707055652724, −4.12868165655432483783533217926, −3.34549200002844314134269220062, −1.02063245602037219351673612806,
1.07181260442366542860995275521, 2.20479134820947487144242474598, 4.03186759175843708530122178623, 5.22042151678595453334649124693, 5.64728356977798912223950837498, 6.73240432258372863425424159283, 7.29277604686392779323532521820, 7.890450447967770847551381878130, 9.401412282394466040154074852731, 10.09238258153788670025141504655