L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (−2.09 − 1.62i)7-s − 0.999·8-s + (2.59 − 1.5i)11-s + 2.44·13-s + (−2.44 + 0.999i)14-s + (−0.5 + 0.866i)16-s + (−0.878 + 0.507i)17-s + (0.878 + 0.507i)19-s − 3i·22-s + (2.12 − 3.67i)23-s + (1.22 − 2.12i)26-s + (−0.358 + 2.62i)28-s + 1.24i·29-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.790 − 0.612i)7-s − 0.353·8-s + (0.783 − 0.452i)11-s + 0.679·13-s + (−0.654 + 0.267i)14-s + (−0.125 + 0.216i)16-s + (−0.213 + 0.123i)17-s + (0.201 + 0.116i)19-s − 0.639i·22-s + (0.442 − 0.766i)23-s + (0.240 − 0.416i)26-s + (−0.0677 + 0.495i)28-s + 0.230i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.916 + 0.401i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.916 + 0.401i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.597304434\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.597304434\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.09 + 1.62i)T \) |
good | 11 | \( 1 + (-2.59 + 1.5i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 2.44T + 13T^{2} \) |
| 17 | \( 1 + (0.878 - 0.507i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.878 - 0.507i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.12 + 3.67i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 1.24iT - 29T^{2} \) |
| 31 | \( 1 + (-4.86 + 2.80i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (7.13 + 4.12i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 2.02T + 41T^{2} \) |
| 43 | \( 1 + 8.24iT - 43T^{2} \) |
| 47 | \( 1 + (-0.878 - 0.507i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.621 - 1.07i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (5.76 + 9.98i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.12 - 2.95i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (8.66 - 5i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 10.2iT - 71T^{2} \) |
| 73 | \( 1 + (4.18 + 7.24i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (5.62 - 9.73i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 3.16iT - 83T^{2} \) |
| 89 | \( 1 + (-5.19 + 9i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 3.76T + 97T^{2} \) |
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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
−8.673407387007682553833587421686, −7.53629839800169053883061833527, −6.62553899960187406467621761604, −6.18795661234426156809141386654, −5.20814026801967413274889079283, −4.17208153399803660364741905208, −3.63247295599865092627920906428, −2.82007384387539619283710943897, −1.54876355757100120027580658059, −0.46025222438845360600323756199,
1.37906134474279920846647837273, 2.77948626662283703340830774911, 3.52069347166313669722670002628, 4.41922321926008724060312315549, 5.29142565877515405797950900733, 6.09320799071587366276894662783, 6.67222309082693005191198851132, 7.29443187609592745955497929368, 8.331346690402843717705919499088, 8.908933900722681695950443088042