| L(s) = 1 | + 3-s + 2·7-s + 9-s − 4·11-s − 13-s − 6·17-s − 6·19-s + 2·21-s − 5·25-s + 27-s + 2·29-s − 6·31-s − 4·33-s − 10·37-s − 39-s + 8·41-s − 12·43-s + 12·47-s − 3·49-s − 6·51-s + 6·53-s − 6·57-s − 2·61-s + 2·63-s − 2·67-s − 8·71-s + 14·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.755·7-s + 1/3·9-s − 1.20·11-s − 0.277·13-s − 1.45·17-s − 1.37·19-s + 0.436·21-s − 25-s + 0.192·27-s + 0.371·29-s − 1.07·31-s − 0.696·33-s − 1.64·37-s − 0.160·39-s + 1.24·41-s − 1.82·43-s + 1.75·47-s − 3/7·49-s − 0.840·51-s + 0.824·53-s − 0.794·57-s − 0.256·61-s + 0.251·63-s − 0.244·67-s − 0.949·71-s + 1.63·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−8.599714556107010168015828498331, −7.83185628144364824075839346084, −7.19647105474376846232786526511, −6.26978093624195133379323081604, −5.24023933433090021312206264186, −4.55430994518146483287078897645, −3.69935903431572690700118729854, −2.41470622971862790399111223380, −1.92964550565057213795123796979, 0,
1.92964550565057213795123796979, 2.41470622971862790399111223380, 3.69935903431572690700118729854, 4.55430994518146483287078897645, 5.24023933433090021312206264186, 6.26978093624195133379323081604, 7.19647105474376846232786526511, 7.83185628144364824075839346084, 8.599714556107010168015828498331
