L(s) = 1 | − 1.69·2-s + 3-s + 0.874·4-s + (−2.22 + 0.217i)5-s − 1.69·6-s + (0.698 + 2.55i)7-s + 1.90·8-s + 9-s + (3.77 − 0.368i)10-s + (3.25 − 0.658i)11-s + 0.874·12-s + 5.21i·13-s + (−1.18 − 4.32i)14-s + (−2.22 + 0.217i)15-s − 4.98·16-s + 3.07i·17-s + ⋯ |
L(s) = 1 | − 1.19·2-s + 0.577·3-s + 0.437·4-s + (−0.995 + 0.0972i)5-s − 0.692·6-s + (0.263 + 0.964i)7-s + 0.674·8-s + 0.333·9-s + (1.19 − 0.116i)10-s + (0.980 − 0.198i)11-s + 0.252·12-s + 1.44i·13-s + (−0.316 − 1.15i)14-s + (−0.574 + 0.0561i)15-s − 1.24·16-s + 0.746i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.534 - 0.844i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.534 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6335855230\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6335855230\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 + (2.22 - 0.217i)T \) |
| 7 | \( 1 + (-0.698 - 2.55i)T \) |
| 11 | \( 1 + (-3.25 + 0.658i)T \) |
good | 2 | \( 1 + 1.69T + 2T^{2} \) |
| 13 | \( 1 - 5.21iT - 13T^{2} \) |
| 17 | \( 1 - 3.07iT - 17T^{2} \) |
| 19 | \( 1 + 4.27T + 19T^{2} \) |
| 23 | \( 1 + 6.74iT - 23T^{2} \) |
| 29 | \( 1 - 0.101iT - 29T^{2} \) |
| 31 | \( 1 + 6.52iT - 31T^{2} \) |
| 37 | \( 1 - 10.8iT - 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 + 6.80T + 43T^{2} \) |
| 47 | \( 1 + 6.68T + 47T^{2} \) |
| 53 | \( 1 - 2.30iT - 53T^{2} \) |
| 59 | \( 1 - 7.18iT - 59T^{2} \) |
| 61 | \( 1 + 4.40T + 61T^{2} \) |
| 67 | \( 1 - 4.25iT - 67T^{2} \) |
| 71 | \( 1 + 5.83T + 71T^{2} \) |
| 73 | \( 1 - 8.09iT - 73T^{2} \) |
| 79 | \( 1 - 14.2iT - 79T^{2} \) |
| 83 | \( 1 + 7.09iT - 83T^{2} \) |
| 89 | \( 1 - 16.3iT - 89T^{2} \) |
| 97 | \( 1 - 2.98T + 97T^{2} \) |
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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
−9.745876680904217131370708174424, −9.055985558128490510756642732021, −8.427071868328207860739655218651, −8.138296589830743926018397315716, −6.92379079222806805443465707171, −6.33242539501353650604198972556, −4.51067234745244512179037946694, −4.09674582992804128321912010051, −2.54067799115209074484522642767, −1.44526091586099585702888888847,
0.42746661013302209921123472244, 1.56119358150963487914156805752, 3.28394926557310788400960036939, 4.10306515531367673746083649899, 5.00722331470287192414349535978, 6.68483808581902756614085586740, 7.61160600374983941225013870034, 7.74859895550461023759657385820, 8.735568896932181728538321791467, 9.329488399436684988784768495833