| L(s) = 1 | − 4·2-s + 16·4-s − 79·7-s − 64·8-s − 150·11-s + 137·13-s + 316·14-s + 256·16-s + 2.03e3·17-s − 1.96e3·19-s + 600·22-s + 1.35e3·23-s − 548·26-s − 1.26e3·28-s + 2.94e3·29-s + 713·31-s − 1.02e3·32-s − 8.13e3·34-s − 3.23e3·37-s + 7.87e3·38-s − 6.56e3·41-s − 1.95e4·43-s − 2.40e3·44-s − 5.40e3·46-s + 2.11e4·47-s − 1.05e4·49-s + 2.19e3·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.609·7-s − 0.353·8-s − 0.373·11-s + 0.224·13-s + 0.430·14-s + 1/4·16-s + 1.70·17-s − 1.25·19-s + 0.264·22-s + 0.532·23-s − 0.158·26-s − 0.304·28-s + 0.650·29-s + 0.133·31-s − 0.176·32-s − 1.20·34-s − 0.388·37-s + 0.884·38-s − 0.609·41-s − 1.61·43-s − 0.186·44-s − 0.376·46-s + 1.39·47-s − 0.628·49-s + 0.112·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{2} T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 79 T + p^{5} T^{2} \) |
| 11 | \( 1 + 150 T + p^{5} T^{2} \) |
| 13 | \( 1 - 137 T + p^{5} T^{2} \) |
| 17 | \( 1 - 2034 T + p^{5} T^{2} \) |
| 19 | \( 1 + 1969 T + p^{5} T^{2} \) |
| 23 | \( 1 - 1350 T + p^{5} T^{2} \) |
| 29 | \( 1 - 2946 T + p^{5} T^{2} \) |
| 31 | \( 1 - 23 p T + p^{5} T^{2} \) |
| 37 | \( 1 + 3238 T + p^{5} T^{2} \) |
| 41 | \( 1 + 6564 T + p^{5} T^{2} \) |
| 43 | \( 1 + 19579 T + p^{5} T^{2} \) |
| 47 | \( 1 - 450 p T + p^{5} T^{2} \) |
| 53 | \( 1 - 25896 T + p^{5} T^{2} \) |
| 59 | \( 1 + 25350 T + p^{5} T^{2} \) |
| 61 | \( 1 - 50615 T + p^{5} T^{2} \) |
| 67 | \( 1 + 22519 T + p^{5} T^{2} \) |
| 71 | \( 1 + 33900 T + p^{5} T^{2} \) |
| 73 | \( 1 - 82442 T + p^{5} T^{2} \) |
| 79 | \( 1 + 81472 T + p^{5} T^{2} \) |
| 83 | \( 1 + 25782 T + p^{5} T^{2} \) |
| 89 | \( 1 + 103728 T + p^{5} T^{2} \) |
| 97 | \( 1 + 57343 T + p^{5} T^{2} \) |
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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
−10.03144338433269576250453919163, −8.872059246651937095061362120108, −8.153012534040581533170526442656, −7.13127836213897546386570591039, −6.24995196057058524543424579248, −5.20669590093626050305508696042, −3.69141932231699448240217405223, −2.64903738440484987290045448261, −1.24678969763968638964924111906, 0,
1.24678969763968638964924111906, 2.64903738440484987290045448261, 3.69141932231699448240217405223, 5.20669590093626050305508696042, 6.24995196057058524543424579248, 7.13127836213897546386570591039, 8.153012534040581533170526442656, 8.872059246651937095061362120108, 10.03144338433269576250453919163
