L(s) = 1 | + (−119. − 226. i)2-s − 3.78e3i·3-s + (−3.69e4 + 5.41e4i)4-s − 4.81e5·5-s + (−8.57e5 + 4.52e5i)6-s + 5.15e5i·7-s + (1.66e7 + 1.91e6i)8-s − 1.43e7·9-s + (5.75e7 + 1.09e8i)10-s + 1.66e8i·11-s + (2.04e8 + 1.40e8i)12-s − 4.92e8·13-s + (1.16e8 − 6.15e7i)14-s + 1.82e9i·15-s + (−1.55e9 − 4.00e9i)16-s + 7.46e9·17-s + ⋯ |
L(s) = 1 | + (−0.466 − 0.884i)2-s − 0.577i·3-s + (−0.564 + 0.825i)4-s − 1.23·5-s + (−0.510 + 0.269i)6-s + 0.0893i·7-s + (0.993 + 0.113i)8-s − 0.333·9-s + (0.575 + 1.09i)10-s + 0.775i·11-s + (0.476 + 0.325i)12-s − 0.604·13-s + (0.0790 − 0.0417i)14-s + 0.711i·15-s + (−0.362 − 0.931i)16-s + 1.06·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.825 + 0.564i)\, \overline{\Lambda}(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & (0.825 + 0.564i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{17}{2})\) |
\(\approx\) |
\(0.820009 - 0.253507i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.820009 - 0.253507i\) |
\(L(9)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (119. + 226. i)T \) |
| 3 | \( 1 + 3.78e3iT \) |
good | 5 | \( 1 + 4.81e5T + 1.52e11T^{2} \) |
| 7 | \( 1 - 5.15e5iT - 3.32e13T^{2} \) |
| 11 | \( 1 - 1.66e8iT - 4.59e16T^{2} \) |
| 13 | \( 1 + 4.92e8T + 6.65e17T^{2} \) |
| 17 | \( 1 - 7.46e9T + 4.86e19T^{2} \) |
| 19 | \( 1 - 3.55e9iT - 2.88e20T^{2} \) |
| 23 | \( 1 + 1.41e11iT - 6.13e21T^{2} \) |
| 29 | \( 1 - 9.07e11T + 2.50e23T^{2} \) |
| 31 | \( 1 - 1.71e11iT - 7.27e23T^{2} \) |
| 37 | \( 1 + 3.92e12T + 1.23e25T^{2} \) |
| 41 | \( 1 - 6.33e12T + 6.37e25T^{2} \) |
| 43 | \( 1 - 1.38e13iT - 1.36e26T^{2} \) |
| 47 | \( 1 - 3.46e13iT - 5.66e26T^{2} \) |
| 53 | \( 1 - 8.06e13T + 3.87e27T^{2} \) |
| 59 | \( 1 - 1.70e14iT - 2.15e28T^{2} \) |
| 61 | \( 1 - 3.04e14T + 3.67e28T^{2} \) |
| 67 | \( 1 - 3.16e14iT - 1.64e29T^{2} \) |
| 71 | \( 1 + 1.80e14iT - 4.16e29T^{2} \) |
| 73 | \( 1 + 1.28e15T + 6.50e29T^{2} \) |
| 79 | \( 1 + 1.91e15iT - 2.30e30T^{2} \) |
| 83 | \( 1 - 2.98e15iT - 5.07e30T^{2} \) |
| 89 | \( 1 + 8.50e14T + 1.54e31T^{2} \) |
| 97 | \( 1 + 1.30e15T + 6.14e31T^{2} \) |
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
−16.28341307146337277143386580377, −14.46148618451809886038046791056, −12.50981379880280013518411300275, −11.93732678338092189875995435371, −10.29244447115550912972938471918, −8.427267393218790076062421787486, −7.30303638010695352938263453658, −4.40881827306690316217908575669, −2.70727303487181833561538748145, −0.855987303459895895914255798529,
0.55950069651480980350766947969, 3.72605900442154651581777272146, 5.34906942686574631778549457582, 7.34496955963939689331477414868, 8.562075554285211680753851591548, 10.16824305542066946533075318283, 11.73400865982265849522249723391, 13.97273183595459424130520722246, 15.31758758362267838735934569132, 16.10537237234048082030069025234