L(s) = 1 | + (0.382 − 0.923i)7-s + (0.382 + 0.923i)9-s + (0.216 + 0.324i)11-s + (0.707 − 0.292i)23-s + (0.923 + 0.382i)25-s + (−1.08 + 1.63i)29-s + (0.216 − 1.08i)37-s + (0.923 − 0.617i)43-s + (−0.707 − 0.707i)49-s + (0.923 + 1.38i)53-s + 63-s + (−1.38 − 0.923i)67-s + (0.541 − 1.30i)71-s + (0.382 − 0.0761i)77-s + (1.30 + 1.30i)79-s + ⋯ |
L(s) = 1 | + (0.382 − 0.923i)7-s + (0.382 + 0.923i)9-s + (0.216 + 0.324i)11-s + (0.707 − 0.292i)23-s + (0.923 + 0.382i)25-s + (−1.08 + 1.63i)29-s + (0.216 − 1.08i)37-s + (0.923 − 0.617i)43-s + (−0.707 − 0.707i)49-s + (0.923 + 1.38i)53-s + 63-s + (−1.38 − 0.923i)67-s + (0.541 − 1.30i)71-s + (0.382 − 0.0761i)77-s + (1.30 + 1.30i)79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.428071335\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.428071335\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-0.382 + 0.923i)T \) |
good | 3 | \( 1 + (-0.382 - 0.923i)T^{2} \) |
| 5 | \( 1 + (-0.923 - 0.382i)T^{2} \) |
| 11 | \( 1 + (-0.216 - 0.324i)T + (-0.382 + 0.923i)T^{2} \) |
| 13 | \( 1 + (-0.923 + 0.382i)T^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 19 | \( 1 + (0.923 - 0.382i)T^{2} \) |
| 23 | \( 1 + (-0.707 + 0.292i)T + (0.707 - 0.707i)T^{2} \) |
| 29 | \( 1 + (1.08 - 1.63i)T + (-0.382 - 0.923i)T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + (-0.216 + 1.08i)T + (-0.923 - 0.382i)T^{2} \) |
| 41 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 43 | \( 1 + (-0.923 + 0.617i)T + (0.382 - 0.923i)T^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 + (-0.923 - 1.38i)T + (-0.382 + 0.923i)T^{2} \) |
| 59 | \( 1 + (-0.923 - 0.382i)T^{2} \) |
| 61 | \( 1 + (-0.382 - 0.923i)T^{2} \) |
| 67 | \( 1 + (1.38 + 0.923i)T + (0.382 + 0.923i)T^{2} \) |
| 71 | \( 1 + (-0.541 + 1.30i)T + (-0.707 - 0.707i)T^{2} \) |
| 73 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 79 | \( 1 + (-1.30 - 1.30i)T + iT^{2} \) |
| 83 | \( 1 + (0.923 - 0.382i)T^{2} \) |
| 89 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−8.836129296315143284447648331670, −7.79162247209442995235692434466, −7.30985523891772222484157705980, −6.79877557685503611690923464811, −5.59055870534455300870535396919, −4.89276953979563794381355748501, −4.22661404294962526313510960421, −3.32162454629884592162181488922, −2.14015438198263714535510167101, −1.17908716964799183519340592230,
1.05763875084802447877328725654, 2.27122566921199880228700095367, 3.18441893669126692677826933961, 4.10059929504666672595762881661, 4.94643208129582265008506812326, 5.83307462936359152132081254850, 6.39076554111867216645333168986, 7.23001982769493958656622920240, 8.079082675234512949329958081312, 8.792550516232014720379164751042