L(s) = 1 | + (1.78 + 1.78i)3-s + (−0.707 + 0.707i)5-s + i·7-s + 3.40i·9-s + (3.20 − 3.20i)11-s + (4.01 + 4.01i)13-s − 2.53·15-s + 2.63·17-s + (1.42 + 1.42i)19-s + (−1.78 + 1.78i)21-s − 7.25i·23-s − 1.00i·25-s + (−0.725 + 0.725i)27-s + (5.26 + 5.26i)29-s + 3.94·31-s + ⋯ |
L(s) = 1 | + (1.03 + 1.03i)3-s + (−0.316 + 0.316i)5-s + 0.377i·7-s + 1.13i·9-s + (0.966 − 0.966i)11-s + (1.11 + 1.11i)13-s − 0.653·15-s + 0.640·17-s + (0.326 + 0.326i)19-s + (−0.390 + 0.390i)21-s − 1.51i·23-s − 0.200i·25-s + (−0.139 + 0.139i)27-s + (0.978 + 0.978i)29-s + 0.708·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0897 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0897 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.815262075\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.815262075\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.707 - 0.707i)T \) |
| 7 | \( 1 - iT \) |
good | 3 | \( 1 + (-1.78 - 1.78i)T + 3iT^{2} \) |
| 11 | \( 1 + (-3.20 + 3.20i)T - 11iT^{2} \) |
| 13 | \( 1 + (-4.01 - 4.01i)T + 13iT^{2} \) |
| 17 | \( 1 - 2.63T + 17T^{2} \) |
| 19 | \( 1 + (-1.42 - 1.42i)T + 19iT^{2} \) |
| 23 | \( 1 + 7.25iT - 23T^{2} \) |
| 29 | \( 1 + (-5.26 - 5.26i)T + 29iT^{2} \) |
| 31 | \( 1 - 3.94T + 31T^{2} \) |
| 37 | \( 1 + (4.55 - 4.55i)T - 37iT^{2} \) |
| 41 | \( 1 + 4.78iT - 41T^{2} \) |
| 43 | \( 1 + (3.44 - 3.44i)T - 43iT^{2} \) |
| 47 | \( 1 + 10.1T + 47T^{2} \) |
| 53 | \( 1 + (3.16 - 3.16i)T - 53iT^{2} \) |
| 59 | \( 1 + (-5.09 + 5.09i)T - 59iT^{2} \) |
| 61 | \( 1 + (9.21 + 9.21i)T + 61iT^{2} \) |
| 67 | \( 1 + (-5.47 - 5.47i)T + 67iT^{2} \) |
| 71 | \( 1 - 13.9iT - 71T^{2} \) |
| 73 | \( 1 - 4.28iT - 73T^{2} \) |
| 79 | \( 1 + 4.27T + 79T^{2} \) |
| 83 | \( 1 + (10.7 + 10.7i)T + 83iT^{2} \) |
| 89 | \( 1 - 0.251iT - 89T^{2} \) |
| 97 | \( 1 + 1.49T + 97T^{2} \) |
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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
−9.010692282882340263832383541953, −8.577309482077511880238385216211, −8.143125575083473252255615237681, −6.69857908259191484295651476322, −6.29551851956855389671801817175, −4.99419671876597607226916065177, −4.12711862378913389282533164206, −3.46699268413729851767420245641, −2.84631008042896503172479842006, −1.38831645622982359491866823500,
1.01773096334295798844862574121, 1.71350639377938130072265183227, 3.07054134286129646211658428442, 3.66174664880008193266973590441, 4.74205454939787557735796527253, 5.88059729762512832499376117536, 6.77581927300742071772066175707, 7.46135200060625148523019262547, 8.047299448999686900032944852923, 8.608129959948751320315610881534