L(s) = 1 | + (2.82 + 2.82i)5-s + 3i·9-s + (2.82 − 2.82i)13-s + 2·17-s + 11.0i·25-s + (−2.82 + 2.82i)29-s + (−8.48 − 8.48i)37-s + 10i·41-s + (−8.48 + 8.48i)45-s + 7·49-s + (−2.82 − 2.82i)53-s + (8.48 − 8.48i)61-s + 16.0·65-s + 6i·73-s − 9·81-s + ⋯ |
L(s) = 1 | + (1.26 + 1.26i)5-s + i·9-s + (0.784 − 0.784i)13-s + 0.485·17-s + 2.20i·25-s + (−0.525 + 0.525i)29-s + (−1.39 − 1.39i)37-s + 1.56i·41-s + (−1.26 + 1.26i)45-s + 49-s + (−0.388 − 0.388i)53-s + (1.08 − 1.08i)61-s + 1.98·65-s + 0.702i·73-s − 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.037042393\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.037042393\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 - 3iT^{2} \) |
| 5 | \( 1 + (-2.82 - 2.82i)T + 5iT^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 11iT^{2} \) |
| 13 | \( 1 + (-2.82 + 2.82i)T - 13iT^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 19iT^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + (2.82 - 2.82i)T - 29iT^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + (8.48 + 8.48i)T + 37iT^{2} \) |
| 41 | \( 1 - 10iT - 41T^{2} \) |
| 43 | \( 1 + 43iT^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + (2.82 + 2.82i)T + 53iT^{2} \) |
| 59 | \( 1 + 59iT^{2} \) |
| 61 | \( 1 + (-8.48 + 8.48i)T - 61iT^{2} \) |
| 67 | \( 1 - 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 83iT^{2} \) |
| 89 | \( 1 - 10iT - 89T^{2} \) |
| 97 | \( 1 + 18T + 97T^{2} \) |
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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
−10.23569053622082086164273179807, −9.483706590324607110963825750688, −8.412932550411099968127786977725, −7.49411578571984833129837680311, −6.69639240747841423715517868433, −5.77145181090535733330274541279, −5.22980056795375191386541686254, −3.60979273820762860950997702900, −2.68017699394703907603747882189, −1.69087459122260463322952877064,
1.00574272082875459696307284961, 1.99892333399940755888540440435, 3.56948708546434868197597579258, 4.59354653671398767727413174612, 5.61539393165454219855814184631, 6.16412383270172508996917318555, 7.14586328455134446352755337421, 8.602773449742420048004251352876, 8.872559472518868588079430028459, 9.703275999072454986569375231145