| L(s) = 1 | − 1.40i·3-s − 1.35·5-s + i·7-s + 1.02·9-s + 1.15i·11-s − 1.66i·13-s + 1.90i·15-s − 2.28i·17-s − 5.03i·19-s + 1.40·21-s + 4.00·23-s − 3.15·25-s − 5.65i·27-s − 2.40i·29-s + 5.53·31-s + ⋯ |
| L(s) = 1 | − 0.810i·3-s − 0.607·5-s + 0.377i·7-s + 0.342·9-s + 0.347i·11-s − 0.461i·13-s + 0.492i·15-s − 0.555i·17-s − 1.15i·19-s + 0.306·21-s + 0.835·23-s − 0.631·25-s − 1.08i·27-s − 0.446i·29-s + 0.994·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.239 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.239 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.292035070\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.292035070\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - iT \) |
| 41 | \( 1 + (-6.21 - 1.53i)T \) |
| good | 3 | \( 1 + 1.40iT - 3T^{2} \) |
| 5 | \( 1 + 1.35T + 5T^{2} \) |
| 11 | \( 1 - 1.15iT - 11T^{2} \) |
| 13 | \( 1 + 1.66iT - 13T^{2} \) |
| 17 | \( 1 + 2.28iT - 17T^{2} \) |
| 19 | \( 1 + 5.03iT - 19T^{2} \) |
| 23 | \( 1 - 4.00T + 23T^{2} \) |
| 29 | \( 1 + 2.40iT - 29T^{2} \) |
| 31 | \( 1 - 5.53T + 31T^{2} \) |
| 37 | \( 1 + 0.838T + 37T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 + 13.4iT - 47T^{2} \) |
| 53 | \( 1 + 6.80iT - 53T^{2} \) |
| 59 | \( 1 + 7.72T + 59T^{2} \) |
| 61 | \( 1 - 3.76T + 61T^{2} \) |
| 67 | \( 1 - 6.05iT - 67T^{2} \) |
| 71 | \( 1 + 13.1iT - 71T^{2} \) |
| 73 | \( 1 - 8.45T + 73T^{2} \) |
| 79 | \( 1 + 1.01iT - 79T^{2} \) |
| 83 | \( 1 + 17.7T + 83T^{2} \) |
| 89 | \( 1 + 7.88iT - 89T^{2} \) |
| 97 | \( 1 - 9.62iT - 97T^{2} \) |
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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
−9.556069064631287515104518547603, −8.576265437799539750089699147800, −7.83101172833729029018946204243, −7.09788713880170207034984488321, −6.47449376114067520498512794699, −5.24623944607198413365672787110, −4.40552513216827047599590948064, −3.14396646216149120727987474872, −2.06685227306495881207096104748, −0.60853621161004042339938793094,
1.41367309132489178605901192361, 3.15147425810233296981412690575, 4.01665056365271009211705894240, 4.59851371102913069034994836252, 5.74730468528206502678002821572, 6.74617603389779810788819601202, 7.66693454993154157849971103568, 8.395036964254120277664885099164, 9.360344658324677986389267567776, 10.04403814537237278055700667173
