L(s) = 1 | + (−0.923 − 0.382i)5-s − i·9-s + (−0.541 + 0.541i)13-s + (1.30 + 1.30i)17-s + (0.707 + 0.707i)25-s − 1.41i·29-s + (1.41 − 1.41i)37-s − 0.765i·41-s + (−0.382 + 0.923i)45-s + (−1 − i)53-s − 1.84i·61-s + (0.707 − 0.292i)65-s + (1.30 − 1.30i)73-s − 81-s + (−0.707 − 1.70i)85-s + ⋯ |
L(s) = 1 | + (−0.923 − 0.382i)5-s − i·9-s + (−0.541 + 0.541i)13-s + (1.30 + 1.30i)17-s + (0.707 + 0.707i)25-s − 1.41i·29-s + (1.41 − 1.41i)37-s − 0.765i·41-s + (−0.382 + 0.923i)45-s + (−1 − i)53-s − 1.84i·61-s + (0.707 − 0.292i)65-s + (1.30 − 1.30i)73-s − 81-s + (−0.707 − 1.70i)85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.450 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.450 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9911110590\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9911110590\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.923 + 0.382i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + iT^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + (0.541 - 0.541i)T - iT^{2} \) |
| 17 | \( 1 + (-1.30 - 1.30i)T + iT^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 + 1.41iT - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + (-1.41 + 1.41i)T - iT^{2} \) |
| 41 | \( 1 + 0.765iT - T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 + (1 + i)T + iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + 1.84iT - T^{2} \) |
| 67 | \( 1 + iT^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (-1.30 + 1.30i)T - iT^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + 1.84T + T^{2} \) |
| 97 | \( 1 + (0.541 + 0.541i)T + iT^{2} \) |
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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
−8.329887479773390888519195784898, −7.908911641255694682267174167440, −7.14497176023244338159429884326, −6.28237680842628693995456060938, −5.61862128700197610987397856645, −4.55815565506652431651530302032, −3.89068674665857923581329180599, −3.28859740290677663932367330546, −1.95123329799602626593511518571, −0.65757041891722221763918326068,
1.15968296798081494997070088521, 2.76501105465121453502921838981, 3.07132741715043310461655784990, 4.32170523990960950344788531879, 4.97704739039762373701516883630, 5.66357704443461337793308674869, 6.81533660551986703766565066997, 7.43905749166730158849059187676, 7.911051005684769804272352562013, 8.543074859996249154624137354886