| L(s) = 1 | − 4.10i·5-s − 7-s − 2.67i·11-s − 3.02i·13-s + 5.12·17-s + 2.78i·19-s + 7.12·23-s − 11.8·25-s − 8.83i·29-s + 1.42·31-s + 4.10i·35-s − 1.42i·37-s − 5.12·41-s + 2.39i·43-s − 9.56·47-s + ⋯ |
| L(s) = 1 | − 1.83i·5-s − 0.377·7-s − 0.807i·11-s − 0.838i·13-s + 1.24·17-s + 0.637i·19-s + 1.48·23-s − 2.36·25-s − 1.63i·29-s + 0.255·31-s + 0.693i·35-s − 0.234i·37-s − 0.800·41-s + 0.365i·43-s − 1.39·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.845 + 0.534i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.845 + 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.413251334\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.413251334\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 + 4.10iT - 5T^{2} \) |
| 11 | \( 1 + 2.67iT - 11T^{2} \) |
| 13 | \( 1 + 3.02iT - 13T^{2} \) |
| 17 | \( 1 - 5.12T + 17T^{2} \) |
| 19 | \( 1 - 2.78iT - 19T^{2} \) |
| 23 | \( 1 - 7.12T + 23T^{2} \) |
| 29 | \( 1 + 8.83iT - 29T^{2} \) |
| 31 | \( 1 - 1.42T + 31T^{2} \) |
| 37 | \( 1 + 1.42iT - 37T^{2} \) |
| 41 | \( 1 + 5.12T + 41T^{2} \) |
| 43 | \( 1 - 2.39iT - 43T^{2} \) |
| 47 | \( 1 + 9.56T + 47T^{2} \) |
| 53 | \( 1 + 2.78iT - 53T^{2} \) |
| 59 | \( 1 + 4iT - 59T^{2} \) |
| 61 | \( 1 - 5.17iT - 61T^{2} \) |
| 67 | \( 1 + 0.244iT - 67T^{2} \) |
| 71 | \( 1 + 4.27T + 71T^{2} \) |
| 73 | \( 1 + 4.15T + 73T^{2} \) |
| 79 | \( 1 + 6.25T + 79T^{2} \) |
| 83 | \( 1 - 9.35iT - 83T^{2} \) |
| 89 | \( 1 + 11.2T + 89T^{2} \) |
| 97 | \( 1 - 6.69T + 97T^{2} \) |
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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
−8.662815369764662792444594262285, −8.245474952936274542139018676818, −7.53721784315922921601249085366, −6.19383335457456163235292430395, −5.52867803878746292554659248533, −4.93641972260132360060843803664, −3.89084120185976094323797370732, −2.99068755226240933115283423841, −1.39266932862749874051297797293, −0.53504506513507596643477098821,
1.68808477764924519034846881587, 2.93688202775279955240539772746, 3.32883985013107710071822543379, 4.56576122548770233578662600640, 5.58211524064320671016781345171, 6.72889807356345266552057897841, 6.90931300863768819892470809121, 7.60458054105692365181203043428, 8.786527188235372227614830458820, 9.653367220446829668102435598511
