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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.232922352\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.232922352\) |
\(L(1)\) |
|
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
−9.256605641591234590716614491715, −8.910795968755229013846123112709, −8.019079866897780880256512781554, −7.01119381271110769742713054870, −6.43072636855418341135310867958, −5.34918744240314911876443062812, −4.73793429289789566293260991401, −3.54206089392569076294207002983, −2.62283527555943498779730812947, −1.35273966865603683291178777174,
1.16337105204809241708173268299, 2.45673267128541477511201942361, 3.57737563456813007123505827355, 4.68293973275078591872219714204, 5.07975479498243589119725502495, 6.42898529256625276954164352132, 7.04198909847934107909991681506, 8.063821145896901604297040894188, 8.301505872208087438308487958892, 9.617300844588676441224855198324