| L(s) = 1 | + 3-s − 5-s + 9-s − 13-s − 15-s + 2·17-s + 4·19-s − 4·23-s + 25-s + 27-s − 6·29-s − 8·31-s − 6·37-s − 39-s − 2·41-s − 4·43-s − 45-s − 7·49-s + 2·51-s + 6·53-s + 4·57-s − 2·61-s + 65-s − 8·67-s − 4·69-s + 6·73-s + 75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.277·13-s − 0.258·15-s + 0.485·17-s + 0.917·19-s − 0.834·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s − 1.43·31-s − 0.986·37-s − 0.160·39-s − 0.312·41-s − 0.609·43-s − 0.149·45-s − 49-s + 0.280·51-s + 0.824·53-s + 0.529·57-s − 0.256·61-s + 0.124·65-s − 0.977·67-s − 0.481·69-s + 0.702·73-s + 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
| 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
−7.57860773190444184507718682655, −7.31410358691720893159017883814, −6.34306412903661323205670846752, −5.44511342280867385521701874991, −4.84243614399591229688061863832, −3.69094427708340770867201211666, −3.46847085138331232413249674831, −2.31242196130868732747290098696, −1.44147343949589411121227002844, 0,
1.44147343949589411121227002844, 2.31242196130868732747290098696, 3.46847085138331232413249674831, 3.69094427708340770867201211666, 4.84243614399591229688061863832, 5.44511342280867385521701874991, 6.34306412903661323205670846752, 7.31410358691720893159017883814, 7.57860773190444184507718682655
