L(s) = 1 | + (1 + i)5-s + 3i·9-s + (−5 + 5i)13-s + 8·17-s − 3i·25-s + (−3 + 3i)29-s + (7 + 7i)37-s + 8i·41-s + (−3 + 3i)45-s + 7·49-s + (−9 − 9i)53-s + (11 − 11i)61-s − 10·65-s − 6i·73-s − 9·81-s + ⋯ |
L(s) = 1 | + (0.447 + 0.447i)5-s + i·9-s + (−1.38 + 1.38i)13-s + 1.94·17-s − 0.600i·25-s + (−0.557 + 0.557i)29-s + (1.15 + 1.15i)37-s + 1.24i·41-s + (−0.447 + 0.447i)45-s + 49-s + (−1.23 − 1.23i)53-s + (1.40 − 1.40i)61-s − 1.24·65-s − 0.702i·73-s − 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.18874 + 0.794291i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.18874 + 0.794291i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 3iT^{2} \) |
| 5 | \( 1 + (-1 - i)T + 5iT^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 11iT^{2} \) |
| 13 | \( 1 + (5 - 5i)T - 13iT^{2} \) |
| 17 | \( 1 - 8T + 17T^{2} \) |
| 19 | \( 1 - 19iT^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + (3 - 3i)T - 29iT^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + (-7 - 7i)T + 37iT^{2} \) |
| 41 | \( 1 - 8iT - 41T^{2} \) |
| 43 | \( 1 + 43iT^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + (9 + 9i)T + 53iT^{2} \) |
| 59 | \( 1 + 59iT^{2} \) |
| 61 | \( 1 + (-11 + 11i)T - 61iT^{2} \) |
| 67 | \( 1 - 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 83iT^{2} \) |
| 89 | \( 1 + 10iT - 89T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
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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
−11.06236775639795567100244089205, −9.896753913495995417310259114373, −9.714459441067935872421169197455, −8.233111594885068046670183703719, −7.45643673119267695566580024035, −6.54334900405889425656547510256, −5.38909910079939966506619296906, −4.51558208219417136938845621403, −2.97598496441294941622438905756, −1.85044050054771805168019821255,
0.892281956705227384142959632513, 2.69885519437037653529796435207, 3.85485476713265904517512018193, 5.36316779682565399323150483515, 5.77660705467302345136684631443, 7.26818589013450089758301648273, 7.908003555302383178461448082083, 9.196400020440862872369552590347, 9.758859936295986518668239176825, 10.52001374860430329285730054128