| L(s) = 1 | − i·2-s − 4-s + 1.03i·5-s + (1.71 − 2.01i)7-s + i·8-s + 1.03·10-s − 4.93·11-s + 4.53·13-s + (−2.01 − 1.71i)14-s + 16-s + 4.03i·17-s + (−3.25 + 2.90i)19-s − 1.03i·20-s + 4.93i·22-s + 1.42·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + 0.461i·5-s + (0.646 − 0.762i)7-s + 0.353i·8-s + 0.326·10-s − 1.48·11-s + 1.25·13-s + (−0.539 − 0.457i)14-s + 0.250·16-s + 0.978i·17-s + (−0.746 + 0.665i)19-s − 0.230i·20-s + 1.05i·22-s + 0.296·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0246i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0246i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.584331912\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.584331912\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1.71 + 2.01i)T \) |
| 19 | \( 1 + (3.25 - 2.90i)T \) |
| good | 5 | \( 1 - 1.03iT - 5T^{2} \) |
| 11 | \( 1 + 4.93T + 11T^{2} \) |
| 13 | \( 1 - 4.53T + 13T^{2} \) |
| 17 | \( 1 - 4.03iT - 17T^{2} \) |
| 23 | \( 1 - 1.42T + 23T^{2} \) |
| 29 | \( 1 - 7.78iT - 29T^{2} \) |
| 31 | \( 1 - 2.76T + 31T^{2} \) |
| 37 | \( 1 + 0.659iT - 37T^{2} \) |
| 41 | \( 1 - 3.30T + 41T^{2} \) |
| 43 | \( 1 + 5.50T + 43T^{2} \) |
| 47 | \( 1 + 11.9iT - 47T^{2} \) |
| 53 | \( 1 - 12.3iT - 53T^{2} \) |
| 59 | \( 1 - 5.06T + 59T^{2} \) |
| 61 | \( 1 + 1.94iT - 61T^{2} \) |
| 67 | \( 1 - 8.44iT - 67T^{2} \) |
| 71 | \( 1 - 2.08iT - 71T^{2} \) |
| 73 | \( 1 - 13.8iT - 73T^{2} \) |
| 79 | \( 1 - 5.34iT - 79T^{2} \) |
| 83 | \( 1 + 4.33iT - 83T^{2} \) |
| 89 | \( 1 - 16.0T + 89T^{2} \) |
| 97 | \( 1 - 13.2T + 97T^{2} \) |
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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
−8.729586821974611538890785855639, −8.405263554274165887437723006868, −7.57255422440129274882165048332, −6.69305531696313611830343857169, −5.71153016812665620555265647210, −4.89021522905528177336694849516, −3.95242174823964378785257475499, −3.23495458759214783165268657090, −2.12630131515660154972018074459, −1.08214825691079690004123870952,
0.63526401000962373230934313769, 2.17807147088010481715634498339, 3.16934789850159191123184883105, 4.62920748244221770500381915645, 4.94047873926502673938627116238, 5.85316584746001426952807327940, 6.51830904515848665521321032144, 7.65381261050879284786521324262, 8.159679795290205735585418267937, 8.782473583452568374359234500957
