| L(s) = 1 | − 3-s − 5-s − 2·7-s + 9-s − 11-s + 15-s + 4·17-s + 4·19-s + 2·21-s − 4·23-s + 25-s − 27-s + 6·29-s + 33-s + 2·35-s − 2·37-s + 2·41-s + 2·43-s − 45-s − 12·47-s − 3·49-s − 4·51-s − 2·53-s + 55-s − 4·57-s − 4·59-s + 2·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s − 0.301·11-s + 0.258·15-s + 0.970·17-s + 0.917·19-s + 0.436·21-s − 0.834·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s + 0.174·33-s + 0.338·35-s − 0.328·37-s + 0.312·41-s + 0.304·43-s − 0.149·45-s − 1.75·47-s − 3/7·49-s − 0.560·51-s − 0.274·53-s + 0.134·55-s − 0.529·57-s − 0.520·59-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.327928248082230186328675054526, −7.71628868509994110289646962435, −6.90835470758380687640993102815, −6.16612681967173480463612841510, −5.41528069520776634370515445572, −4.58168632302442688326468148552, −3.57605740507444799325548971772, −2.84175954669628532536409257901, −1.30766337550682033617411731645, 0,
1.30766337550682033617411731645, 2.84175954669628532536409257901, 3.57605740507444799325548971772, 4.58168632302442688326468148552, 5.41528069520776634370515445572, 6.16612681967173480463612841510, 6.90835470758380687640993102815, 7.71628868509994110289646962435, 8.327928248082230186328675054526
