L(s) = 1 | − 0.261·2-s + (−0.733 − 1.56i)3-s − 1.93·4-s + i·5-s + (0.192 + 0.411i)6-s + i·7-s + 1.03·8-s + (−1.92 + 2.30i)9-s − 0.261i·10-s + (−3.25 + 0.656i)11-s + (1.41 + 3.03i)12-s − 3.73i·13-s − 0.261i·14-s + (1.56 − 0.733i)15-s + 3.59·16-s − 1.46·17-s + ⋯ |
L(s) = 1 | − 0.185·2-s + (−0.423 − 0.905i)3-s − 0.965·4-s + 0.447i·5-s + (0.0784 + 0.167i)6-s + 0.377i·7-s + 0.364·8-s + (−0.641 + 0.767i)9-s − 0.0828i·10-s + (−0.980 + 0.197i)11-s + (0.408 + 0.874i)12-s − 1.03i·13-s − 0.0700i·14-s + (0.405 − 0.189i)15-s + 0.898·16-s − 0.355·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.594 + 0.804i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.594 + 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7448883555\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7448883555\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.733 + 1.56i)T \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 + (3.25 - 0.656i)T \) |
good | 2 | \( 1 + 0.261T + 2T^{2} \) |
| 13 | \( 1 + 3.73iT - 13T^{2} \) |
| 17 | \( 1 + 1.46T + 17T^{2} \) |
| 19 | \( 1 - 5.82iT - 19T^{2} \) |
| 23 | \( 1 + 2.98iT - 23T^{2} \) |
| 29 | \( 1 - 4.36T + 29T^{2} \) |
| 31 | \( 1 - 5.20T + 31T^{2} \) |
| 37 | \( 1 - 2.14T + 37T^{2} \) |
| 41 | \( 1 - 3.91T + 41T^{2} \) |
| 43 | \( 1 - 1.80iT - 43T^{2} \) |
| 47 | \( 1 + 8.56iT - 47T^{2} \) |
| 53 | \( 1 + 14.0iT - 53T^{2} \) |
| 59 | \( 1 + 4.17iT - 59T^{2} \) |
| 61 | \( 1 - 0.658iT - 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 - 0.170iT - 71T^{2} \) |
| 73 | \( 1 - 7.57iT - 73T^{2} \) |
| 79 | \( 1 + 12.2iT - 79T^{2} \) |
| 83 | \( 1 + 5.56T + 83T^{2} \) |
| 89 | \( 1 + 7.47iT - 89T^{2} \) |
| 97 | \( 1 - 9.21T + 97T^{2} \) |
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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
−9.944980420948084098048728281569, −8.381090458789075916955870171389, −8.256060337466597879063588015681, −7.32824498739807133667340436924, −6.23628621599578625732065266246, −5.49881093338496000661912061671, −4.71033859845563401954101513255, −3.27953600126810027626147783089, −2.17424182325188688763964919154, −0.57598065963181983029219518204,
0.802087500802892246848143455099, 2.84245775599550754878592448671, 4.17650172697940532246814300997, 4.62494505043436968925770120281, 5.39208874527061156530267019629, 6.43513253058721651568415611910, 7.62556544634444293453934168526, 8.569595365862186188688960036856, 9.205074569753749499814126751808, 9.766972682571013548530281693823