L(s) = 1 | − 1.41i·3-s + 1.41i·5-s + 1.41i·7-s − 1.00·9-s + 11-s + 13-s + 2.00·15-s − 17-s + 2.00·21-s − 23-s − 1.00·25-s − 1.41i·29-s − 31-s − 1.41i·33-s − 2.00·35-s + ⋯ |
L(s) = 1 | − 1.41i·3-s + 1.41i·5-s + 1.41i·7-s − 1.00·9-s + 11-s + 13-s + 2.00·15-s − 17-s + 2.00·21-s − 23-s − 1.00·25-s − 1.41i·29-s − 31-s − 1.41i·33-s − 2.00·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9786525593\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9786525593\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 43 | \( 1 - T \) |
good | 3 | \( 1 + 1.41iT - T^{2} \) |
| 5 | \( 1 - 1.41iT - T^{2} \) |
| 7 | \( 1 - 1.41iT - T^{2} \) |
| 11 | \( 1 - T + T^{2} \) |
| 13 | \( 1 - T + T^{2} \) |
| 17 | \( 1 + T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + T + T^{2} \) |
| 29 | \( 1 + 1.41iT - T^{2} \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T + T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 + 1.41iT - T^{2} \) |
| 67 | \( 1 - T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 1.41iT - T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + T + T^{2} \) |
| 89 | \( 1 + 1.41iT - T^{2} \) |
| 97 | \( 1 + T + T^{2} \) |
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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
−11.14586790916016265516689251948, −9.722828184949493458412804671018, −8.803275580580571098796398991565, −7.968714389940138550767023084849, −7.01829307296405286244763720686, −6.23104066337324199229623543726, −5.94906825582127718332404478479, −3.94533175139833457262303902305, −2.63069012597740277611429321776, −1.88991669419546720793511996389,
1.30829460957297612815184621612, 3.73801372314853691622162466875, 4.13426754701765094979493895953, 4.88133244660417220198383502627, 6.01865936654350890632958234742, 7.22427689109333868453980504710, 8.475366008458092203203974965334, 9.083773644300897629635361320168, 9.689150343166217424017537918692, 10.76402798893695840830830758613