L(s) = 1 | + (1.41 + i)3-s + (1.73 − 1.41i)5-s + (1.00 + 2.82i)9-s + 4.89·11-s − 4.89i·13-s + (3.86 − 0.267i)15-s − 3.46·17-s − 3.46i·19-s + 6i·23-s + (0.999 − 4.89i)25-s + (−1.41 + 5.00i)27-s + 2.82i·29-s − 3.46i·31-s + (6.92 + 4.89i)33-s + 4.89i·37-s + ⋯ |
L(s) = 1 | + (0.816 + 0.577i)3-s + (0.774 − 0.632i)5-s + (0.333 + 0.942i)9-s + 1.47·11-s − 1.35i·13-s + (0.997 − 0.0691i)15-s − 0.840·17-s − 0.794i·19-s + 1.25i·23-s + (0.199 − 0.979i)25-s + (−0.272 + 0.962i)27-s + 0.525i·29-s − 0.622i·31-s + (1.20 + 0.852i)33-s + 0.805i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0691i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0691i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.54514 + 0.0881473i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.54514 + 0.0881473i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.41 - i)T \) |
| 5 | \( 1 + (-1.73 + 1.41i)T \) |
good | 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 - 4.89T + 11T^{2} \) |
| 13 | \( 1 + 4.89iT - 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 + 3.46iT - 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 - 2.82iT - 29T^{2} \) |
| 31 | \( 1 + 3.46iT - 31T^{2} \) |
| 37 | \( 1 - 4.89iT - 37T^{2} \) |
| 41 | \( 1 + 5.65iT - 41T^{2} \) |
| 43 | \( 1 - 8.48T + 43T^{2} \) |
| 47 | \( 1 - 6iT - 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 - 4.89T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 8.48T + 67T^{2} \) |
| 71 | \( 1 - 9.79T + 71T^{2} \) |
| 73 | \( 1 - 9.79iT - 73T^{2} \) |
| 79 | \( 1 - 10.3iT - 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 - 5.65iT - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−9.709265356310864495015056189015, −9.287459736505532438049453850675, −8.610561109615001691559391804061, −7.71377984332650204444860598032, −6.60619469906005137098814698394, −5.56638560196007685153401736972, −4.69443686020797140175932704918, −3.73119983478107147034021217367, −2.61531076316282649256860049527, −1.34965695404763486023126385774,
1.54596618378507389878858806884, 2.33686649266122610980055544545, 3.59887123424009616905464741030, 4.48054938233500237122820336905, 6.29277029788560230984926995944, 6.47086379209714516901450959578, 7.34053952808881983712941768488, 8.525736622423112062703914048019, 9.203184261154842272957759565002, 9.713200937578747573130447673497