L(s) = 1 | − i·2-s + 3-s − 4-s + i·5-s − i·6-s − 1.04i·7-s + i·8-s + 9-s + 10-s + 1.35i·11-s − 12-s − 1.04·14-s + i·15-s + 16-s + 1.08·17-s − i·18-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + 0.577·3-s − 0.5·4-s + 0.447i·5-s − 0.408i·6-s − 0.396i·7-s + 0.353i·8-s + 0.333·9-s + 0.316·10-s + 0.409i·11-s − 0.288·12-s − 0.280·14-s + 0.258i·15-s + 0.250·16-s + 0.263·17-s − 0.235i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.246 + 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.246 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.197547949\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.197547949\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 - iT \) |
| 13 | \( 1 \) |
good | 7 | \( 1 + 1.04iT - 7T^{2} \) |
| 11 | \( 1 - 1.35iT - 11T^{2} \) |
| 17 | \( 1 - 1.08T + 17T^{2} \) |
| 19 | \( 1 + 2.93iT - 19T^{2} \) |
| 23 | \( 1 - 0.692T + 23T^{2} \) |
| 29 | \( 1 + 2.37T + 29T^{2} \) |
| 31 | \( 1 + 9.85iT - 31T^{2} \) |
| 37 | \( 1 - 9.26iT - 37T^{2} \) |
| 41 | \( 1 + 2.84iT - 41T^{2} \) |
| 43 | \( 1 - 4.45T + 43T^{2} \) |
| 47 | \( 1 + 3.31iT - 47T^{2} \) |
| 53 | \( 1 - 0.664T + 53T^{2} \) |
| 59 | \( 1 + 1.96iT - 59T^{2} \) |
| 61 | \( 1 - 3.24T + 61T^{2} \) |
| 67 | \( 1 + 6.91iT - 67T^{2} \) |
| 71 | \( 1 + 2.29iT - 71T^{2} \) |
| 73 | \( 1 - 3.36iT - 73T^{2} \) |
| 79 | \( 1 - 12.3T + 79T^{2} \) |
| 83 | \( 1 + 2.68iT - 83T^{2} \) |
| 89 | \( 1 - 12.8iT - 89T^{2} \) |
| 97 | \( 1 + 1.56iT - 97T^{2} \) |
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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
−8.088278355982780434670976487012, −7.47490005696833169587669592550, −6.80058029742885716567314301670, −5.88381541891375702072103277668, −4.90995403321821502733567423257, −4.15536834985752522509107247936, −3.47060824740088653738025373561, −2.61923479626675522665413858568, −1.90919581652141770248987512144, −0.66528649261232363227082986245,
0.958950686238257417767813443757, 2.09800867694781227141524271937, 3.20592907274405068141026307224, 3.92236728126149788115255034819, 4.81315908795850007339255859877, 5.57310768039566642331932778741, 6.13212407366705482298130353389, 7.13029237105071750140592693794, 7.65040244103542356089429805092, 8.469946666195637611634585866847