L(s) = 1 | + (−2 + i)5-s − 4i·13-s + 2i·17-s + (3 − 4i)25-s − 4·29-s − 12i·37-s + 8·41-s + 7·49-s − 14i·53-s + 10·61-s + (4 + 8i)65-s − 16i·73-s + (−2 − 4i)85-s + 16·89-s − 8i·97-s + ⋯ |
L(s) = 1 | + (−0.894 + 0.447i)5-s − 1.10i·13-s + 0.485i·17-s + (0.600 − 0.800i)25-s − 0.742·29-s − 1.97i·37-s + 1.24·41-s + 49-s − 1.92i·53-s + 1.28·61-s + (0.496 + 0.992i)65-s − 1.87i·73-s + (−0.216 − 0.433i)85-s + 1.69·89-s − 0.812i·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.087861922\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.087861922\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2 - i)T \) |
good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 12iT - 37T^{2} \) |
| 41 | \( 1 - 8T + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 14iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 16iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 16T + 89T^{2} \) |
| 97 | \( 1 + 8iT - 97T^{2} \) |
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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
−9.357567053701169161926616503547, −8.453381622514453753375831803736, −7.73887529773438920537607144060, −7.14792364725153639479603561366, −6.09046397723231810391365551442, −5.26633385497965809184663136883, −4.08541305420778551649624752280, −3.41091975851096070929084056928, −2.27971455438168325708404875579, −0.50473177986945510754361832342,
1.15705028952187128664649683041, 2.61159662032030680773899388489, 3.83470254322702316279446150214, 4.49710116028780179666916488972, 5.41502213200377348833706332231, 6.54525525749204763048290778765, 7.31092479107903217698039894091, 8.050081986763397716150557060162, 8.925301369306121627365714418917, 9.458679988768253084379139758445