| L(s) = 1 | − 2-s + 4-s − 5-s − 1.56·7-s − 8-s + 10-s + 1.56·11-s + 2·13-s + 1.56·14-s + 16-s − 5.12·17-s + 4.68·19-s − 20-s − 1.56·22-s − 5.56·23-s + 25-s − 2·26-s − 1.56·28-s + 1.12·29-s + 31-s − 32-s + 5.12·34-s + 1.56·35-s + 5.12·37-s − 4.68·38-s + 40-s + 1.12·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.447·5-s − 0.590·7-s − 0.353·8-s + 0.316·10-s + 0.470·11-s + 0.554·13-s + 0.417·14-s + 0.250·16-s − 1.24·17-s + 1.07·19-s − 0.223·20-s − 0.332·22-s − 1.15·23-s + 0.200·25-s − 0.392·26-s − 0.295·28-s + 0.208·29-s + 0.179·31-s − 0.176·32-s + 0.878·34-s + 0.263·35-s + 0.842·37-s − 0.759·38-s + 0.158·40-s + 0.175·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 + 1.56T + 7T^{2} \) |
| 11 | \( 1 - 1.56T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 5.12T + 17T^{2} \) |
| 19 | \( 1 - 4.68T + 19T^{2} \) |
| 23 | \( 1 + 5.56T + 23T^{2} \) |
| 29 | \( 1 - 1.12T + 29T^{2} \) |
| 37 | \( 1 - 5.12T + 37T^{2} \) |
| 41 | \( 1 - 1.12T + 41T^{2} \) |
| 43 | \( 1 + 7.80T + 43T^{2} \) |
| 47 | \( 1 - 3.12T + 47T^{2} \) |
| 53 | \( 1 + 11.5T + 53T^{2} \) |
| 59 | \( 1 - 4.87T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 9.36T + 67T^{2} \) |
| 71 | \( 1 + 4.68T + 71T^{2} \) |
| 73 | \( 1 - 9.80T + 73T^{2} \) |
| 79 | \( 1 + 16.6T + 79T^{2} \) |
| 83 | \( 1 - 2.24T + 83T^{2} \) |
| 89 | \( 1 - 1.31T + 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−8.354743832392681783151520635946, −7.889559608806654526613142391429, −6.80986850271746098711925464653, −6.47159808835076253954481404234, −5.47505764581384809578360766360, −4.31215541623833777994529899236, −3.53523513964495048951777514468, −2.55556984015940836652716599214, −1.33302563099350497200222708063, 0,
1.33302563099350497200222708063, 2.55556984015940836652716599214, 3.53523513964495048951777514468, 4.31215541623833777994529899236, 5.47505764581384809578360766360, 6.47159808835076253954481404234, 6.80986850271746098711925464653, 7.889559608806654526613142391429, 8.354743832392681783151520635946
