L(s) = 1 | + (0.195 + 0.980i)2-s + (−0.923 + 0.382i)4-s + (0.555 − 0.831i)7-s + (−0.555 − 0.831i)8-s + (0.831 − 0.555i)9-s + (0.598 + 1.11i)11-s + (0.923 + 0.382i)14-s + (0.707 − 0.707i)16-s + (0.707 + 0.707i)18-s + (−0.980 + 0.804i)22-s + (−0.324 + 1.63i)23-s + (0.980 − 0.195i)25-s + (−0.195 + 0.980i)28-s + (−1.75 − 0.938i)29-s + (0.831 + 0.555i)32-s + ⋯ |
L(s) = 1 | + (0.195 + 0.980i)2-s + (−0.923 + 0.382i)4-s + (0.555 − 0.831i)7-s + (−0.555 − 0.831i)8-s + (0.831 − 0.555i)9-s + (0.598 + 1.11i)11-s + (0.923 + 0.382i)14-s + (0.707 − 0.707i)16-s + (0.707 + 0.707i)18-s + (−0.980 + 0.804i)22-s + (−0.324 + 1.63i)23-s + (0.980 − 0.195i)25-s + (−0.195 + 0.980i)28-s + (−1.75 − 0.938i)29-s + (0.831 + 0.555i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.427 - 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.427 - 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.117482229\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.117482229\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.195 - 0.980i)T \) |
| 7 | \( 1 + (-0.555 + 0.831i)T \) |
good | 3 | \( 1 + (-0.831 + 0.555i)T^{2} \) |
| 5 | \( 1 + (-0.980 + 0.195i)T^{2} \) |
| 11 | \( 1 + (-0.598 - 1.11i)T + (-0.555 + 0.831i)T^{2} \) |
| 13 | \( 1 + (0.980 + 0.195i)T^{2} \) |
| 17 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 19 | \( 1 + (0.195 - 0.980i)T^{2} \) |
| 23 | \( 1 + (0.324 - 1.63i)T + (-0.923 - 0.382i)T^{2} \) |
| 29 | \( 1 + (1.75 + 0.938i)T + (0.555 + 0.831i)T^{2} \) |
| 31 | \( 1 + iT^{2} \) |
| 37 | \( 1 + (-0.728 - 0.598i)T + (0.195 + 0.980i)T^{2} \) |
| 41 | \( 1 + (-0.923 - 0.382i)T^{2} \) |
| 43 | \( 1 + (1.68 + 0.512i)T + (0.831 + 0.555i)T^{2} \) |
| 47 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 53 | \( 1 + (-0.512 + 0.273i)T + (0.555 - 0.831i)T^{2} \) |
| 59 | \( 1 + (0.980 - 0.195i)T^{2} \) |
| 61 | \( 1 + (0.831 - 0.555i)T^{2} \) |
| 67 | \( 1 + (-0.555 - 1.83i)T + (-0.831 + 0.555i)T^{2} \) |
| 71 | \( 1 + (0.636 + 0.425i)T + (0.382 + 0.923i)T^{2} \) |
| 73 | \( 1 + (0.382 - 0.923i)T^{2} \) |
| 79 | \( 1 + (0.750 + 1.81i)T + (-0.707 + 0.707i)T^{2} \) |
| 83 | \( 1 + (-0.195 + 0.980i)T^{2} \) |
| 89 | \( 1 + (0.923 - 0.382i)T^{2} \) |
| 97 | \( 1 + iT^{2} \) |
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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
−10.02832865065181842903218598726, −9.694161032125574604539887425821, −8.693078466816705710100639137481, −7.52897176906051554964727834006, −7.24204844253952868329906609338, −6.34451969150137062668504555226, −5.16992068272891935662794895069, −4.28138766898870443097603704225, −3.65870363918694437442290120750, −1.50682452866242756158153507527,
1.45448663612628308723069310551, 2.56912683068754057524288460001, 3.73812044465765558586863221424, 4.75334125186112925992808748180, 5.53218078785755375523299200915, 6.57967067287108050890386722995, 7.954654871475566169326766489187, 8.714104614896546331478529975297, 9.326854495596192874891519327873, 10.40236874488143752584347722947