L(s) = 1 | + (−1.84 − 1.84i)3-s + (−2.41 + 2.41i)5-s + 1.53i·7-s + 3.82i·9-s + (3.37 − 3.37i)11-s + (−0.414 − 0.414i)13-s + 8.92·15-s + 2.82·17-s + (−0.317 − 0.317i)19-s + (2.82 − 2.82i)21-s − 5.86i·23-s − 6.65i·25-s + (1.53 − 1.53i)27-s + (3.24 + 3.24i)29-s + 7.39·31-s + ⋯ |
L(s) = 1 | + (−1.06 − 1.06i)3-s + (−1.07 + 1.07i)5-s + 0.578i·7-s + 1.27i·9-s + (1.01 − 1.01i)11-s + (−0.114 − 0.114i)13-s + 2.30·15-s + 0.685·17-s + (−0.0727 − 0.0727i)19-s + (0.617 − 0.617i)21-s − 1.22i·23-s − 1.33i·25-s + (0.294 − 0.294i)27-s + (0.602 + 0.602i)29-s + 1.32·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.741653 - 0.307202i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.741653 - 0.307202i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + (1.84 + 1.84i)T + 3iT^{2} \) |
| 5 | \( 1 + (2.41 - 2.41i)T - 5iT^{2} \) |
| 7 | \( 1 - 1.53iT - 7T^{2} \) |
| 11 | \( 1 + (-3.37 + 3.37i)T - 11iT^{2} \) |
| 13 | \( 1 + (0.414 + 0.414i)T + 13iT^{2} \) |
| 17 | \( 1 - 2.82T + 17T^{2} \) |
| 19 | \( 1 + (0.317 + 0.317i)T + 19iT^{2} \) |
| 23 | \( 1 + 5.86iT - 23T^{2} \) |
| 29 | \( 1 + (-3.24 - 3.24i)T + 29iT^{2} \) |
| 31 | \( 1 - 7.39T + 31T^{2} \) |
| 37 | \( 1 + (-3.58 + 3.58i)T - 37iT^{2} \) |
| 41 | \( 1 + 4iT - 41T^{2} \) |
| 43 | \( 1 + (-1.84 + 1.84i)T - 43iT^{2} \) |
| 47 | \( 1 + 7.39T + 47T^{2} \) |
| 53 | \( 1 + (-5.24 + 5.24i)T - 53iT^{2} \) |
| 59 | \( 1 + (-1.84 + 1.84i)T - 59iT^{2} \) |
| 61 | \( 1 + (-9.24 - 9.24i)T + 61iT^{2} \) |
| 67 | \( 1 + (-7.07 - 7.07i)T + 67iT^{2} \) |
| 71 | \( 1 + 11.9iT - 71T^{2} \) |
| 73 | \( 1 - 10.4iT - 73T^{2} \) |
| 79 | \( 1 + 6.12T + 79T^{2} \) |
| 83 | \( 1 + (-2.48 - 2.48i)T + 83iT^{2} \) |
| 89 | \( 1 - 0.828iT - 89T^{2} \) |
| 97 | \( 1 - 10.8T + 97T^{2} \) |
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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
−11.09025727704089331387092617445, −10.28530396532919245288654395289, −8.725754600728912890557623952735, −7.908354740016928367820687213340, −6.86458392852709368555143361383, −6.44482912306864472181137527250, −5.45668505040692680068795208708, −3.91982951127300207157823204613, −2.69082056999746400524263862751, −0.77224484747695618626007886078,
1.00125856704785019149684505453, 3.74053036532520893199862754555, 4.42179928323541649983103938674, 5.00224154552871222080299473178, 6.25345657465302009314041871773, 7.41775109531669985229256353186, 8.358055459108206850481128885664, 9.610305720737028485098219358945, 9.950928362730505717124441256552, 11.18930109990846122966743062715