| L(s) = 1 | − 3-s + (−2 + i)5-s − i·7-s + 9-s − 6·13-s + (2 − i)15-s − 2i·17-s − 4i·19-s + i·21-s + 4i·23-s + (3 − 4i)25-s − 27-s − 6i·29-s + 8·31-s + (1 + 2i)35-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + (−0.894 + 0.447i)5-s − 0.377i·7-s + 0.333·9-s − 1.66·13-s + (0.516 − 0.258i)15-s − 0.485i·17-s − 0.917i·19-s + 0.218i·21-s + 0.834i·23-s + (0.600 − 0.800i)25-s − 0.192·27-s − 1.11i·29-s + 1.43·31-s + (0.169 + 0.338i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.316 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.316 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6316687074\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6316687074\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + (2 - i)T \) |
| 7 | \( 1 + iT \) |
| good | 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 6T + 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 8T + 41T^{2} \) |
| 43 | \( 1 - 6T + 43T^{2} \) |
| 47 | \( 1 - 2iT - 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 6iT - 59T^{2} \) |
| 61 | \( 1 - 10iT - 61T^{2} \) |
| 67 | \( 1 - 2T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−8.682811177184510242258009697753, −7.77841685468538486035892709426, −7.23329015721359118440342809506, −6.76772507938308963759707337596, −5.73419146841517276423694697831, −4.73957362977125156160147748847, −4.37000814100310825777023089899, −3.19149967919829601241504471599, −2.38518871903012722702557574717, −0.74833576553058149841603470573,
0.30728108029786993346128404027, 1.69732792631369941281081976029, 2.89560237545425143389815825411, 3.89895878746919370239977757652, 4.82702048847931642479359662314, 5.16868697802840566543630991686, 6.28699624342385626985278406297, 6.96567477981761374923079124435, 7.80531120093154186454057323951, 8.330930547511987971438085556120
