L(s) = 1 | + 3-s + 2·5-s + 9-s − 6·13-s + 2·15-s − 6·17-s − 19-s − 4·23-s − 25-s + 27-s − 2·29-s − 8·31-s + 10·37-s − 6·39-s − 2·41-s − 4·43-s + 2·45-s − 12·47-s − 7·49-s − 6·51-s + 6·53-s − 57-s − 12·59-s + 2·61-s − 12·65-s − 4·67-s − 4·69-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.894·5-s + 1/3·9-s − 1.66·13-s + 0.516·15-s − 1.45·17-s − 0.229·19-s − 0.834·23-s − 1/5·25-s + 0.192·27-s − 0.371·29-s − 1.43·31-s + 1.64·37-s − 0.960·39-s − 0.312·41-s − 0.609·43-s + 0.298·45-s − 1.75·47-s − 49-s − 0.840·51-s + 0.824·53-s − 0.132·57-s − 1.56·59-s + 0.256·61-s − 1.48·65-s − 0.488·67-s − 0.481·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 19 | \( 1 + T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−8.131011445127830248766373318777, −7.50211428031250118028389448361, −6.68366663778337482202254497130, −6.02118792891233657053500601741, −5.02650326286180649130370068752, −4.42654993025529810423971512307, −3.33953886441583183350883619745, −2.23486091264922987348941709400, −1.92863015818052613281166898081, 0,
1.92863015818052613281166898081, 2.23486091264922987348941709400, 3.33953886441583183350883619745, 4.42654993025529810423971512307, 5.02650326286180649130370068752, 6.02118792891233657053500601741, 6.68366663778337482202254497130, 7.50211428031250118028389448361, 8.131011445127830248766373318777