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
−8.543074859996249154624137354886, −7.911051005684769804272352562013, −7.43905749166730158849059187676, −6.81533660551986703766565066997, −5.66357704443461337793308674869, −4.97704739039762373701516883630, −4.32170523990960950344788531879, −3.07132741715043310461655784990, −2.76501105465121453502921838981, −1.15968296798081494997070088521,
0.65757041891722221763918326068, 1.95123329799602626593511518571, 3.28859740290677663932367330546, 3.89068674665857923581329180599, 4.55815565506652431651530302032, 5.61862128700197610987397856645, 6.28237680842628693995456060938, 7.14497176023244338159429884326, 7.908911641255694682267174167440, 8.329887479773390888519195784898