L(s) = 1 | + (−1.22 − 1.22i)3-s + (2.44 − 2.44i)7-s + 2.99i·9-s + 6·11-s + (−12.2 − 12.2i)13-s + (14.6 − 14.6i)17-s + 10i·19-s − 5.99·21-s + (−29.3 − 29.3i)23-s + (3.67 − 3.67i)27-s − 48i·29-s − 26·31-s + (−7.34 − 7.34i)33-s + (31.8 − 31.8i)37-s + 29.9i·39-s + ⋯ |
L(s) = 1 | + (−0.408 − 0.408i)3-s + (0.349 − 0.349i)7-s + 0.333i·9-s + 0.545·11-s + (−0.942 − 0.942i)13-s + (0.864 − 0.864i)17-s + 0.526i·19-s − 0.285·21-s + (−1.27 − 1.27i)23-s + (0.136 − 0.136i)27-s − 1.65i·29-s − 0.838·31-s + (−0.222 − 0.222i)33-s + (0.860 − 0.860i)37-s + 0.769i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.731451 - 0.924228i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.731451 - 0.924228i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.22 + 1.22i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-2.44 + 2.44i)T - 49iT^{2} \) |
| 11 | \( 1 - 6T + 121T^{2} \) |
| 13 | \( 1 + (12.2 + 12.2i)T + 169iT^{2} \) |
| 17 | \( 1 + (-14.6 + 14.6i)T - 289iT^{2} \) |
| 19 | \( 1 - 10iT - 361T^{2} \) |
| 23 | \( 1 + (29.3 + 29.3i)T + 529iT^{2} \) |
| 29 | \( 1 + 48iT - 841T^{2} \) |
| 31 | \( 1 + 26T + 961T^{2} \) |
| 37 | \( 1 + (-31.8 + 31.8i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 - 30T + 1.68e3T^{2} \) |
| 43 | \( 1 + (29.3 + 29.3i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (14.6 - 14.6i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (-14.6 - 14.6i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + 78iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 2T + 3.72e3T^{2} \) |
| 67 | \( 1 + (63.6 - 63.6i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 - 120T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-83.2 - 83.2i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 - 74iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-44.0 - 44.0i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + 150iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-4.89 + 4.89i)T - 9.40e3iT^{2} \) |
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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
−11.37669514483362885454861803666, −10.30643632625457843569425696771, −9.571373964782326649407953125629, −8.060899960733864964473059238848, −7.51293964782241922817181356554, −6.27079342999003316445655415580, −5.28427629352129898941472488007, −4.05119257295352495427827487243, −2.37571800585112722676551406369, −0.61418325216137052688984252443,
1.72947024362064091859072374161, 3.54518201069897943392201660301, 4.73113548586262852139873328416, 5.73219737959411815104379795015, 6.84939025934657416715551721536, 7.967667284387947838045391143910, 9.170840792551643709967694071449, 9.835052444124533417539444781145, 10.94187598126337078591804278233, 11.82376057234325607329569936964