L(s) = 1 | − 1.29i·2-s + i·3-s + 0.335·4-s + i·5-s + 1.29·6-s + (−2.23 + 1.42i)7-s − 3.01i·8-s − 9-s + 1.29·10-s + (0.810 + 3.21i)11-s + 0.335i·12-s − 3.24·13-s + (1.83 + 2.87i)14-s − 15-s − 3.21·16-s − 7.53·17-s + ⋯ |
L(s) = 1 | − 0.912i·2-s + 0.577i·3-s + 0.167·4-s + 0.447i·5-s + 0.526·6-s + (−0.843 + 0.537i)7-s − 1.06i·8-s − 0.333·9-s + 0.408·10-s + (0.244 + 0.969i)11-s + 0.0967i·12-s − 0.898·13-s + (0.490 + 0.769i)14-s − 0.258·15-s − 0.804·16-s − 1.82·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.727 - 0.686i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.727 - 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3265907599\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3265907599\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.23 - 1.42i)T \) |
| 11 | \( 1 + (-0.810 - 3.21i)T \) |
good | 2 | \( 1 + 1.29iT - 2T^{2} \) |
| 13 | \( 1 + 3.24T + 13T^{2} \) |
| 17 | \( 1 + 7.53T + 17T^{2} \) |
| 19 | \( 1 + 2.10T + 19T^{2} \) |
| 23 | \( 1 + 1.54T + 23T^{2} \) |
| 29 | \( 1 + 7.87iT - 29T^{2} \) |
| 31 | \( 1 + 2.42iT - 31T^{2} \) |
| 37 | \( 1 + 8.46T + 37T^{2} \) |
| 41 | \( 1 - 6.82T + 41T^{2} \) |
| 43 | \( 1 - 8.43iT - 43T^{2} \) |
| 47 | \( 1 + 2.05iT - 47T^{2} \) |
| 53 | \( 1 - 11.0T + 53T^{2} \) |
| 59 | \( 1 - 9.43iT - 59T^{2} \) |
| 61 | \( 1 + 8.80T + 61T^{2} \) |
| 67 | \( 1 + 11.0T + 67T^{2} \) |
| 71 | \( 1 + 11.2T + 71T^{2} \) |
| 73 | \( 1 - 3.80T + 73T^{2} \) |
| 79 | \( 1 - 5.78iT - 79T^{2} \) |
| 83 | \( 1 - 9.24T + 83T^{2} \) |
| 89 | \( 1 - 13.1iT - 89T^{2} \) |
| 97 | \( 1 - 2.85iT - 97T^{2} \) |
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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
−10.18785098112319697531225649663, −9.514401964994435216824728612292, −8.941009049472157416079241295039, −7.51127332182069352729137725738, −6.69805829668709898159501406288, −6.03297739826114635755589401526, −4.56896410612117719048373628857, −3.89998468305799371095107737350, −2.61631727221126745040328340900, −2.20766575424191718536806264197,
0.12334063532624017913988116253, 1.94446488466589469286033375211, 3.12217736120499351298457404154, 4.45457626661471090119259743532, 5.50976215775096962256532179972, 6.36171974877182622577693063016, 6.93808071551708958037497384688, 7.55236499026857046501005877946, 8.807704323232801428436393035763, 8.882353043349813531960764272508