L(s) = 1 | + 1.25·3-s + 2.55i·5-s − i·7-s − 1.43·9-s − 0.347i·11-s + (−3.40 + 1.19i)13-s + 3.20i·15-s + 2.78·17-s + 6.54i·19-s − 1.25i·21-s + 6.22·23-s − 1.54·25-s − 5.54·27-s − 7.72·29-s + 7.29i·31-s + ⋯ |
L(s) = 1 | + 0.722·3-s + 1.14i·5-s − 0.377i·7-s − 0.478·9-s − 0.104i·11-s + (−0.943 + 0.330i)13-s + 0.826i·15-s + 0.674·17-s + 1.50i·19-s − 0.272i·21-s + 1.29·23-s − 0.308·25-s − 1.06·27-s − 1.43·29-s + 1.30i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.330 - 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.330 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.566694371\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.566694371\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + iT \) |
| 13 | \( 1 + (3.40 - 1.19i)T \) |
good | 3 | \( 1 - 1.25T + 3T^{2} \) |
| 5 | \( 1 - 2.55iT - 5T^{2} \) |
| 11 | \( 1 + 0.347iT - 11T^{2} \) |
| 17 | \( 1 - 2.78T + 17T^{2} \) |
| 19 | \( 1 - 6.54iT - 19T^{2} \) |
| 23 | \( 1 - 6.22T + 23T^{2} \) |
| 29 | \( 1 + 7.72T + 29T^{2} \) |
| 31 | \( 1 - 7.29iT - 31T^{2} \) |
| 37 | \( 1 - 6.07iT - 37T^{2} \) |
| 41 | \( 1 - 5.60iT - 41T^{2} \) |
| 43 | \( 1 - 6.77T + 43T^{2} \) |
| 47 | \( 1 - 8.92iT - 47T^{2} \) |
| 53 | \( 1 + 5.64T + 53T^{2} \) |
| 59 | \( 1 - 0.320iT - 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 + 8.16iT - 67T^{2} \) |
| 71 | \( 1 + 2.83iT - 71T^{2} \) |
| 73 | \( 1 + 10.7iT - 73T^{2} \) |
| 79 | \( 1 - 3.70T + 79T^{2} \) |
| 83 | \( 1 + 13.5iT - 83T^{2} \) |
| 89 | \( 1 - 7.10iT - 89T^{2} \) |
| 97 | \( 1 - 5.45iT - 97T^{2} \) |
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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
−9.667275357187317002424731849952, −9.086127104637893186991650576192, −7.908314560034284507393089783853, −7.54391358492512914464914747271, −6.62115893233437202911823063195, −5.75865246977208070896502053543, −4.63569487011891062510731814664, −3.29555518885534367191984676204, −3.04400790351919465406914393499, −1.70754984842966094673087481069,
0.54743125758605164811450155104, 2.15320519806493621706762517233, 2.99158021068783988850968499347, 4.18115898765752179358279196722, 5.20874452480799377757737547384, 5.63902421649601911891287088402, 7.12566177047041798487189079598, 7.74664445339523752371747263926, 8.681834601396866526947005415379, 9.174723521720203005343742991990