| 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
−9.491973591340646048042156450092, −8.935755426926913113299254390339, −7.990932435138363939780233207117, −7.20193359568108377064226640052, −6.39586698756903199113613682761, −5.64313805054734520611209319778, −4.39490075694743580899427563742, −3.31812950262203798077953644909, −2.63796297977678640697064765347, −1.67297525242083758931082385915,
0.36610869730299114080534750203, 1.26563327947287026317789640942, 2.55941111097199569215934546874, 3.77294956684868207837356296936, 4.47708149159309960115864715617, 5.32321978035018856059151985427, 6.48460458531797366932811701444, 7.38325361791443631494914326821, 8.148806507243685641561214023592, 8.792703139402043228553601433459
