| L(s) = 1 | − 2.23i·5-s + 4.95i·7-s + 10.4i·11-s − 12.6·13-s − 22.4i·17-s − 11.3i·19-s + (−22.4 − 5.15i)23-s − 5.00·25-s + 40.2·29-s − 11.9·31-s + 11.0·35-s − 13.6i·37-s + 24.9·41-s − 8.14i·43-s − 31.5·47-s + ⋯ |
| L(s) = 1 | − 0.447i·5-s + 0.707i·7-s + 0.951i·11-s − 0.973·13-s − 1.32i·17-s − 0.594i·19-s + (−0.974 − 0.224i)23-s − 0.200·25-s + 1.38·29-s − 0.384·31-s + 0.316·35-s − 0.369i·37-s + 0.609·41-s − 0.189i·43-s − 0.670·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.224 - 0.974i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.224 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.336965701\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.336965701\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 2.23iT \) |
| 23 | \( 1 + (22.4 + 5.15i)T \) |
| good | 7 | \( 1 - 4.95iT - 49T^{2} \) |
| 11 | \( 1 - 10.4iT - 121T^{2} \) |
| 13 | \( 1 + 12.6T + 169T^{2} \) |
| 17 | \( 1 + 22.4iT - 289T^{2} \) |
| 19 | \( 1 + 11.3iT - 361T^{2} \) |
| 29 | \( 1 - 40.2T + 841T^{2} \) |
| 31 | \( 1 + 11.9T + 961T^{2} \) |
| 37 | \( 1 + 13.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 24.9T + 1.68e3T^{2} \) |
| 43 | \( 1 + 8.14iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 31.5T + 2.20e3T^{2} \) |
| 53 | \( 1 - 53.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 99.1T + 3.48e3T^{2} \) |
| 61 | \( 1 + 88.9iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 126. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 54.6T + 5.04e3T^{2} \) |
| 73 | \( 1 - 98.8T + 5.32e3T^{2} \) |
| 79 | \( 1 - 47.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 38.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 56.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 83.0iT - 9.40e3T^{2} \) |
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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
−8.484709570976343054695557994504, −7.59121024476493666017695753696, −7.05955634300100465123632210298, −6.21058710764337570417276843721, −5.18970654136405460321314466563, −4.85615449229663057592161385827, −3.96659385532869346530338000710, −2.61767293283380570484850210890, −2.24937416792641713430702937100, −0.836823869935728145215994826491,
0.33768876885198143383221017560, 1.54425089943258047929387862696, 2.58811035009887416318877635224, 3.55865272427941638868552012799, 4.12453334235570813658132709893, 5.13348910405615021831988149931, 6.03585728499823611941324251691, 6.53990913155057454128399930062, 7.44311701877081241113271069734, 8.055730853589451272405688293952
