| L(s) = 1 | + 3-s + 5-s + 7-s + 9-s + 11-s − 13-s + 15-s − 5·17-s − 2·19-s + 21-s − 7·23-s + 25-s + 27-s − 4·31-s + 33-s + 35-s − 7·37-s − 39-s − 11·41-s − 6·43-s + 45-s − 6·49-s − 5·51-s + 11·53-s + 55-s − 2·57-s − 4·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.301·11-s − 0.277·13-s + 0.258·15-s − 1.21·17-s − 0.458·19-s + 0.218·21-s − 1.45·23-s + 1/5·25-s + 0.192·27-s − 0.718·31-s + 0.174·33-s + 0.169·35-s − 1.15·37-s − 0.160·39-s − 1.71·41-s − 0.914·43-s + 0.149·45-s − 6/7·49-s − 0.700·51-s + 1.51·53-s + 0.134·55-s − 0.264·57-s − 0.520·59-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 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−7.79529273088278907784013356192, −6.91323883359094720890722632364, −6.45071753632865669141717017700, −5.51360514405621568072532567397, −4.74268231304202681015889646363, −4.02288909399056598814308505897, −3.20323336097535815009044204940, −2.08993886314282002727781727744, −1.71679282412490868196836997213, 0,
1.71679282412490868196836997213, 2.08993886314282002727781727744, 3.20323336097535815009044204940, 4.02288909399056598814308505897, 4.74268231304202681015889646363, 5.51360514405621568072532567397, 6.45071753632865669141717017700, 6.91323883359094720890722632364, 7.79529273088278907784013356192
