L(s) = 1 | + (−1.29 − 2.24i)5-s + (−0.5 + 0.866i)7-s + (−2.25 + 3.90i)11-s + (−0.5 − 0.866i)13-s + 0.945·17-s + 4.05·19-s + (0.136 + 0.236i)23-s + (−0.863 + 1.49i)25-s + (1.23 − 2.13i)29-s + (1.16 + 2.01i)31-s + 2.59·35-s + 1.78·37-s + (−3.20 − 5.54i)41-s + (−5.21 + 9.03i)43-s + (6.08 − 10.5i)47-s + ⋯ |
L(s) = 1 | + (−0.579 − 1.00i)5-s + (−0.188 + 0.327i)7-s + (−0.680 + 1.17i)11-s + (−0.138 − 0.240i)13-s + 0.229·17-s + 0.930·19-s + (0.0284 + 0.0493i)23-s + (−0.172 + 0.299i)25-s + (0.228 − 0.395i)29-s + (0.209 + 0.362i)31-s + 0.438·35-s + 0.292·37-s + (−0.500 − 0.866i)41-s + (−0.795 + 1.37i)43-s + (0.887 − 1.53i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.384 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.384 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.255470363\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.255470363\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.5 - 0.866i)T \) |
good | 5 | \( 1 + (1.29 + 2.24i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (2.25 - 3.90i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.5 + 0.866i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 0.945T + 17T^{2} \) |
| 19 | \( 1 - 4.05T + 19T^{2} \) |
| 23 | \( 1 + (-0.136 - 0.236i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.23 + 2.13i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.16 - 2.01i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 1.78T + 37T^{2} \) |
| 41 | \( 1 + (3.20 + 5.54i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (5.21 - 9.03i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-6.08 + 10.5i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 6.27T + 53T^{2} \) |
| 59 | \( 1 + (-1.36 - 2.36i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.13 + 1.96i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (7.90 + 13.6i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 3.27T + 71T^{2} \) |
| 73 | \( 1 + 1.50T + 73T^{2} \) |
| 79 | \( 1 + (-7.35 + 12.7i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.472 + 0.819i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 14.3T + 89T^{2} \) |
| 97 | \( 1 + (-5.74 + 9.95i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.572011774009834665519132173640, −7.80358728421328715998789239908, −7.31777242305725886986648361761, −6.29309917232794836068101172038, −5.20015059918443407890053922416, −4.89807717871471860510735748971, −3.92776839640958451897583665166, −2.90770674311356464086694725163, −1.81554668926605472739836447786, −0.50596554448087163993437101572,
0.923179596162115159141153386716, 2.57448810091211612169585302298, 3.24040134576854126308159665591, 3.92682316270510290491657835480, 5.08030138036105643499190916127, 5.87183416806325939587692257884, 6.72131093846265036284308165087, 7.37521563954284378676847867656, 7.984541077151626232897580016566, 8.763001827180421620881202161338