| L(s) = 1 | + 3-s + 2·5-s + 7-s + 9-s + 2·13-s + 2·15-s + 2·17-s − 4·19-s + 21-s − 25-s + 27-s + 6·29-s + 2·35-s + 6·37-s + 2·39-s − 6·41-s − 8·43-s + 2·45-s − 8·47-s + 49-s + 2·51-s + 6·53-s − 4·57-s + 12·59-s + 10·61-s + 63-s + 4·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 0.377·7-s + 1/3·9-s + 0.554·13-s + 0.516·15-s + 0.485·17-s − 0.917·19-s + 0.218·21-s − 1/5·25-s + 0.192·27-s + 1.11·29-s + 0.338·35-s + 0.986·37-s + 0.320·39-s − 0.937·41-s − 1.21·43-s + 0.298·45-s − 1.16·47-s + 1/7·49-s + 0.280·51-s + 0.824·53-s − 0.529·57-s + 1.56·59-s + 1.28·61-s + 0.125·63-s + 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.228005814\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.228005814\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−10.24733695220424559147521672427, −9.768998369578198086360697721326, −8.643952701675766415492574439921, −8.166047761688637687090227543209, −6.91666929474613726367436300689, −6.06997282501662098824886849744, −5.03856124527433249174894736338, −3.89046750453116338353014311196, −2.62233033257522210046559530213, −1.50798537152368882588814373343,
1.50798537152368882588814373343, 2.62233033257522210046559530213, 3.89046750453116338353014311196, 5.03856124527433249174894736338, 6.06997282501662098824886849744, 6.91666929474613726367436300689, 8.166047761688637687090227543209, 8.643952701675766415492574439921, 9.768998369578198086360697721326, 10.24733695220424559147521672427
