L(s) = 1 | + 3·3-s − 12.7i·5-s + (−2.54 − 18.3i)7-s + 9·9-s + 2.42i·11-s − 26.6i·13-s − 38.1i·15-s − 10.6i·17-s − 30.0·19-s + (−7.62 − 55.0i)21-s − 120. i·23-s − 36.7·25-s + 27·27-s − 43.1·29-s + 6.30·31-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 1.13i·5-s + (−0.137 − 0.990i)7-s + 0.333·9-s + 0.0664i·11-s − 0.568i·13-s − 0.656i·15-s − 0.151i·17-s − 0.363·19-s + (−0.0792 − 0.571i)21-s − 1.08i·23-s − 0.293·25-s + 0.192·27-s − 0.276·29-s + 0.0365·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.137i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.990 + 0.137i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.749685557\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.749685557\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 7 | \( 1 + (2.54 + 18.3i)T \) |
good | 5 | \( 1 + 12.7iT - 125T^{2} \) |
| 11 | \( 1 - 2.42iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 26.6iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 10.6iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 30.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 120. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 43.1T + 2.43e4T^{2} \) |
| 31 | \( 1 - 6.30T + 2.97e4T^{2} \) |
| 37 | \( 1 - 61.6T + 5.06e4T^{2} \) |
| 41 | \( 1 + 75.9iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 212. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 330.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 4.87T + 1.48e5T^{2} \) |
| 59 | \( 1 + 481.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 236. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 661. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 547. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 968. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 64.6iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 665.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 384. iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 1.22e3iT - 9.12e5T^{2} \) |
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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
−8.792703139402043228553601433459, −8.148806507243685641561214023592, −7.38325361791443631494914326821, −6.48460458531797366932811701444, −5.32321978035018856059151985427, −4.47708149159309960115864715617, −3.77294956684868207837356296936, −2.55941111097199569215934546874, −1.26563327947287026317789640942, −0.36610869730299114080534750203,
1.67297525242083758931082385915, 2.63796297977678640697064765347, 3.31812950262203798077953644909, 4.39490075694743580899427563742, 5.64313805054734520611209319778, 6.39586698756903199113613682761, 7.20193359568108377064226640052, 7.990932435138363939780233207117, 8.935755426926913113299254390339, 9.491973591340646048042156450092