L(s) = 1 | + 2.27i·3-s + 4.53i·7-s − 2.16·9-s + 6.13·13-s + 8.23·17-s + 4.35i·19-s − 10.3·21-s − 2.86i·23-s + 5·25-s + 1.90i·27-s − 10.6·29-s − 8.71·37-s + 13.9i·39-s − 6i·47-s − 13.5·49-s + ⋯ |
L(s) = 1 | + 1.31i·3-s + 1.71i·7-s − 0.721·9-s + 1.70·13-s + 1.99·17-s + 0.999i·19-s − 2.24·21-s − 0.596i·23-s + 25-s + 0.365i·27-s − 1.98·29-s − 1.43·37-s + 2.23i·39-s − 0.875i·47-s − 1.93·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.812183733\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.812183733\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 - 4.35iT \) |
good | 3 | \( 1 - 2.27iT - 3T^{2} \) |
| 5 | \( 1 - 5T^{2} \) |
| 7 | \( 1 - 4.53iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 6.13T + 13T^{2} \) |
| 17 | \( 1 - 8.23T + 17T^{2} \) |
| 23 | \( 1 + 2.86iT - 23T^{2} \) |
| 29 | \( 1 + 10.6T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 8.71T + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 6iT - 47T^{2} \) |
| 53 | \( 1 + 2.95T + 53T^{2} \) |
| 59 | \( 1 + 14.5iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 5.44iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 15.6T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−9.959504149547157642915157542780, −9.232948300642814999252332688385, −8.638629013733774280933063823047, −7.920721536851764556932490704807, −6.41140979217804061323318487696, −5.52627515693596945779892259123, −5.20190142582109717149918718320, −3.68843535605050506794327344571, −3.31312460625448148879951640577, −1.73112876918739201559841519292,
0.902363341602461692473437675316, 1.47906090322538787854137960859, 3.24743475384073925113687777647, 3.96793445864777943370344335218, 5.36551815793862251478044758222, 6.30359997117693088734792549716, 7.17789144359820282130489966047, 7.51786776187290472538472034500, 8.342229691596337895063183955525, 9.381996715458224787635908422540