L(s) = 1 | + (−0.382 + 0.923i)7-s + (−0.382 − 0.923i)9-s + (1.63 − 1.08i)11-s + (0.707 − 0.292i)23-s + (−0.923 − 0.382i)25-s + (0.324 + 0.216i)29-s + (1.63 + 0.324i)37-s + (0.923 + 1.38i)43-s + (−0.707 − 0.707i)49-s + (0.923 − 0.617i)53-s + 63-s + (0.617 − 0.923i)67-s + (−0.541 + 1.30i)71-s + (0.382 + 1.92i)77-s + (−1.30 − 1.30i)79-s + ⋯ |
L(s) = 1 | + (−0.382 + 0.923i)7-s + (−0.382 − 0.923i)9-s + (1.63 − 1.08i)11-s + (0.707 − 0.292i)23-s + (−0.923 − 0.382i)25-s + (0.324 + 0.216i)29-s + (1.63 + 0.324i)37-s + (0.923 + 1.38i)43-s + (−0.707 − 0.707i)49-s + (0.923 − 0.617i)53-s + 63-s + (0.617 − 0.923i)67-s + (−0.541 + 1.30i)71-s + (0.382 + 1.92i)77-s + (−1.30 − 1.30i)79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 + 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 + 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.161558002\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.161558002\) |
\(L(1)\) |
|
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 | \( 1 + (-1.63 + 1.08i)T + (0.382 - 0.923i)T^{2} \) |
| 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 | \( 1 + (-0.324 - 0.216i)T + (0.382 + 0.923i)T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + (-1.63 - 0.324i)T + (0.923 + 0.382i)T^{2} \) |
| 41 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 43 | \( 1 + (-0.923 - 1.38i)T + (-0.382 + 0.923i)T^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 + (-0.923 + 0.617i)T + (0.382 - 0.923i)T^{2} \) |
| 59 | \( 1 + (0.923 + 0.382i)T^{2} \) |
| 61 | \( 1 + (0.382 + 0.923i)T^{2} \) |
| 67 | \( 1 + (-0.617 + 0.923i)T + (-0.382 - 0.923i)T^{2} \) |
| 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
−9.249981920833421041381667447467, −8.844232442038079942687742337025, −8.073529029115961106450433140040, −6.77203546130765159114518771094, −6.20113256823655667553079997485, −5.70231136970826123312882161984, −4.34318542016166654093871642496, −3.47249608988627339979921558485, −2.65546481801332640547621893441, −1.07440512623192178225494739388,
1.33490287060637897343076651866, 2.52620547517382661211322628620, 3.87812391465162544664653689493, 4.35241890283861609589443543987, 5.47082460797607461617844902079, 6.43872085052859933609468504443, 7.21932238641797150788291584663, 7.72018436645580173359276164721, 8.859887245696887159935920121430, 9.532195988687775325419346078309