L(s) = 1 | − 0.517i·2-s − i·3-s + 1.73·4-s + i·5-s − 0.517·6-s + (−2.63 + 0.189i)7-s − 1.93i·8-s − 9-s + 0.517·10-s + (−3 − 1.41i)11-s − 1.73i·12-s − 4.24·13-s + (0.0980 + 1.36i)14-s + 15-s + 2.46·16-s − 2.44·17-s + ⋯ |
L(s) = 1 | − 0.366i·2-s − 0.577i·3-s + 0.866·4-s + 0.447i·5-s − 0.211·6-s + (−0.997 + 0.0716i)7-s − 0.683i·8-s − 0.333·9-s + 0.163·10-s + (−0.904 − 0.426i)11-s − 0.499i·12-s − 1.17·13-s + (0.0262 + 0.365i)14-s + 0.258·15-s + 0.616·16-s − 0.594·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.932 - 0.360i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.932 - 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3961812008\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3961812008\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + iT \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 + (2.63 - 0.189i)T \) |
| 11 | \( 1 + (3 + 1.41i)T \) |
good | 2 | \( 1 + 0.517iT - 2T^{2} \) |
| 13 | \( 1 + 4.24T + 13T^{2} \) |
| 17 | \( 1 + 2.44T + 17T^{2} \) |
| 19 | \( 1 + 1.03T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 1.03iT - 29T^{2} \) |
| 31 | \( 1 - 2.92iT - 31T^{2} \) |
| 37 | \( 1 + 0.535T + 37T^{2} \) |
| 41 | \( 1 + 6.69T + 41T^{2} \) |
| 43 | \( 1 + 2.44iT - 43T^{2} \) |
| 47 | \( 1 + 2.92iT - 47T^{2} \) |
| 53 | \( 1 + 3.46T + 53T^{2} \) |
| 59 | \( 1 + 12.9iT - 59T^{2} \) |
| 61 | \( 1 - 4.89T + 61T^{2} \) |
| 67 | \( 1 + 5.46T + 67T^{2} \) |
| 71 | \( 1 - 6.92T + 71T^{2} \) |
| 73 | \( 1 - 0.656T + 73T^{2} \) |
| 79 | \( 1 - 3.86iT - 79T^{2} \) |
| 83 | \( 1 + 1.41T + 83T^{2} \) |
| 89 | \( 1 + 4.92iT - 89T^{2} \) |
| 97 | \( 1 - 4.92iT - 97T^{2} \) |
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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
−9.618350422522458675660054827776, −8.377331545023135709798040113331, −7.51477269261616758135699163982, −6.80402890213382304888971945985, −6.23215341289863885720514434124, −5.21313447104757542612532939805, −3.63666741946224978290655273762, −2.74825601567449766374974249917, −2.07588439921134512039498218431, −0.14550037017156548309862704724,
2.16601855167766652144854309280, 3.00391756598629546721638850963, 4.29955435964012619566024578971, 5.23558218595587904418892456438, 6.05009753004828892983540368475, 6.95648851557187701184091028284, 7.68368469293301790408760213874, 8.568407779550971827418567999056, 9.624835682546161987273568601621, 10.12585631382004387660765471455