L(s) = 1 | + (−2.98 − 0.323i)3-s − 2.23i·5-s − 4.72·7-s + (8.79 + 1.92i)9-s + 4.76i·11-s − 1.06·13-s + (−0.722 + 6.66i)15-s + 26.7i·17-s + 8.12·19-s + (14.0 + 1.52i)21-s + 40.0i·23-s − 5.00·25-s + (−25.5 − 8.59i)27-s + 20.8i·29-s + 33.7·31-s + ⋯ |
L(s) = 1 | + (−0.994 − 0.107i)3-s − 0.447i·5-s − 0.675·7-s + (0.976 + 0.214i)9-s + 0.433i·11-s − 0.0820·13-s + (−0.0481 + 0.444i)15-s + 1.57i·17-s + 0.427·19-s + (0.671 + 0.0727i)21-s + 1.74i·23-s − 0.200·25-s + (−0.948 − 0.318i)27-s + 0.719i·29-s + 1.08·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.107 - 0.994i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.107 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.543885 + 0.488133i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.543885 + 0.488133i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.98 + 0.323i)T \) |
| 5 | \( 1 + 2.23iT \) |
good | 7 | \( 1 + 4.72T + 49T^{2} \) |
| 11 | \( 1 - 4.76iT - 121T^{2} \) |
| 13 | \( 1 + 1.06T + 169T^{2} \) |
| 17 | \( 1 - 26.7iT - 289T^{2} \) |
| 19 | \( 1 - 8.12T + 361T^{2} \) |
| 23 | \( 1 - 40.0iT - 529T^{2} \) |
| 29 | \( 1 - 20.8iT - 841T^{2} \) |
| 31 | \( 1 - 33.7T + 961T^{2} \) |
| 37 | \( 1 + 60.4T + 1.36e3T^{2} \) |
| 41 | \( 1 - 59.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 56.4T + 1.84e3T^{2} \) |
| 47 | \( 1 - 9.68iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 93.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 17.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 57.7T + 3.72e3T^{2} \) |
| 67 | \( 1 + 101.T + 4.48e3T^{2} \) |
| 71 | \( 1 + 90.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 40.0T + 5.32e3T^{2} \) |
| 79 | \( 1 + 65.3T + 6.24e3T^{2} \) |
| 83 | \( 1 - 117. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 119. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 15.2T + 9.40e3T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.21737358804047088902199462993, −11.24792588189457332344826285482, −10.20485856271300827976027342433, −9.479709508132708697221792353859, −8.096579210158151059709374160567, −6.94976008475916709814890434364, −5.98368982327554382364459590289, −4.98675239712568564609822213831, −3.67273605486859833904148968647, −1.48069572165790512402447122684,
0.46329095058666196335320256745, 2.82138135405087754376863975133, 4.34986499965077986003149051122, 5.58276059629886240077691448216, 6.57626856104183754628313021378, 7.36264669320283454110883353926, 8.935916322910389379569400470097, 10.00536071008361504494156418275, 10.68942209226179438033700951371, 11.73759790442830963160443387886