L(s) = 1 | + (0.923 + 0.382i)2-s + (0.707 + 0.707i)4-s + (0.382 − 0.923i)7-s + (0.382 + 0.923i)8-s + (−0.382 − 0.923i)9-s + (−1.63 + 1.08i)11-s + (0.707 − 0.707i)14-s + i·16-s − i·18-s + (−1.92 + 0.382i)22-s + (−0.707 + 0.292i)23-s + (−0.923 − 0.382i)25-s + (0.923 − 0.382i)28-s + (0.324 + 0.216i)29-s + (−0.382 + 0.923i)32-s + ⋯ |
L(s) = 1 | + (0.923 + 0.382i)2-s + (0.707 + 0.707i)4-s + (0.382 − 0.923i)7-s + (0.382 + 0.923i)8-s + (−0.382 − 0.923i)9-s + (−1.63 + 1.08i)11-s + (0.707 − 0.707i)14-s + i·16-s − i·18-s + (−1.92 + 0.382i)22-s + (−0.707 + 0.292i)23-s + (−0.923 − 0.382i)25-s + (0.923 − 0.382i)28-s + (0.324 + 0.216i)29-s + (−0.382 + 0.923i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.348362145\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.348362145\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.923 - 0.382i)T \) |
| 7 | \( 1 + (-0.382 + 0.923i)T \) |
good | 3 | \( 1 + (0.382 + 0.923i)T^{2} \) |
| 5 | \( 1 + (0.923 + 0.382i)T^{2} \) |
| 11 | \( 1 + (1.63 - 1.08i)T + (0.382 - 0.923i)T^{2} \) |
| 13 | \( 1 + (0.923 - 0.382i)T^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 19 | \( 1 + (-0.923 + 0.382i)T^{2} \) |
| 23 | \( 1 + (0.707 - 0.292i)T + (0.707 - 0.707i)T^{2} \) |
| 29 | \( 1 + (-0.324 - 0.216i)T + (0.382 + 0.923i)T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + (-1.63 - 0.324i)T + (0.923 + 0.382i)T^{2} \) |
| 41 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 43 | \( 1 + (0.923 + 1.38i)T + (-0.382 + 0.923i)T^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 + (-0.923 + 0.617i)T + (0.382 - 0.923i)T^{2} \) |
| 59 | \( 1 + (0.923 + 0.382i)T^{2} \) |
| 61 | \( 1 + (0.382 + 0.923i)T^{2} \) |
| 67 | \( 1 + (0.617 - 0.923i)T + (-0.382 - 0.923i)T^{2} \) |
| 71 | \( 1 + (-0.541 + 1.30i)T + (-0.707 - 0.707i)T^{2} \) |
| 73 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 79 | \( 1 + (-1.30 - 1.30i)T + iT^{2} \) |
| 83 | \( 1 + (-0.923 + 0.382i)T^{2} \) |
| 89 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−11.61228442457266119951403210222, −10.57333326349591098139342442887, −9.808936546442561759807368362338, −8.251689162258001549592734496789, −7.60475445671023742231489118347, −6.72214542204593961241395153067, −5.60546482070183577005311446829, −4.63911563441289088642065602172, −3.66916154241648517205810318620, −2.30170510873943331393055199239,
2.21115274458941097455014775859, 3.01039583882603933707881056376, 4.60794267644005067742792906239, 5.52780161358421626177057858936, 6.03750220180130946964951952416, 7.73143666031816254928227354276, 8.300174201075096143275329910796, 9.704933223970745721241592491119, 10.72074279127834943229143714834, 11.25889680765901997333165693436