| L(s) = 1 | + 3-s + 9-s − 2·11-s − 13-s + 2·17-s − 5·19-s + 6·23-s − 5·25-s + 27-s − 8·29-s + 3·31-s − 2·33-s − 9·37-s − 39-s − 2·41-s + 43-s − 8·47-s + 2·51-s + 6·53-s − 5·57-s − 6·59-s + 2·61-s − 5·67-s + 6·69-s + 4·71-s + 11·73-s − 5·75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 0.603·11-s − 0.277·13-s + 0.485·17-s − 1.14·19-s + 1.25·23-s − 25-s + 0.192·27-s − 1.48·29-s + 0.538·31-s − 0.348·33-s − 1.47·37-s − 0.160·39-s − 0.312·41-s + 0.152·43-s − 1.16·47-s + 0.280·51-s + 0.824·53-s − 0.662·57-s − 0.781·59-s + 0.256·61-s − 0.610·67-s + 0.722·69-s + 0.474·71-s + 1.28·73-s − 0.577·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.012960075352623823386411092040, −7.27927124440344201104274344928, −6.64767122748064502728044043177, −5.64654135482352039772704253473, −5.00476697518548308574680942920, −4.07140155186937957274708742162, −3.30947762744870335665862903249, −2.43326376011529312910969376107, −1.56818327349053259501041908634, 0,
1.56818327349053259501041908634, 2.43326376011529312910969376107, 3.30947762744870335665862903249, 4.07140155186937957274708742162, 5.00476697518548308574680942920, 5.64654135482352039772704253473, 6.64767122748064502728044043177, 7.27927124440344201104274344928, 8.012960075352623823386411092040
