L(s) = 1 | + 3-s + 9-s + 2·11-s − 4·13-s + 27-s + 2·29-s − 10·31-s + 2·33-s − 4·37-s − 4·39-s + 6·41-s − 4·43-s − 7·49-s − 2·53-s + 6·59-s + 6·61-s − 4·67-s − 8·71-s − 14·73-s − 2·79-s + 81-s − 4·83-s + 2·87-s + 14·89-s − 10·93-s + 6·97-s + 2·99-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1/3·9-s + 0.603·11-s − 1.10·13-s + 0.192·27-s + 0.371·29-s − 1.79·31-s + 0.348·33-s − 0.657·37-s − 0.640·39-s + 0.937·41-s − 0.609·43-s − 49-s − 0.274·53-s + 0.781·59-s + 0.768·61-s − 0.488·67-s − 0.949·71-s − 1.63·73-s − 0.225·79-s + 1/9·81-s − 0.439·83-s + 0.214·87-s + 1.48·89-s − 1.03·93-s + 0.609·97-s + 0.201·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 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
−7.27711862987395826528025364459, −6.90914754016952047170564110259, −5.99037067063811904637463777581, −5.22730767949500544179590445797, −4.50753475643451906610697064345, −3.77250606559791997398842448097, −3.02784900762770343060351748140, −2.19720835436163515574507704553, −1.39406431045706120001293063819, 0,
1.39406431045706120001293063819, 2.19720835436163515574507704553, 3.02784900762770343060351748140, 3.77250606559791997398842448097, 4.50753475643451906610697064345, 5.22730767949500544179590445797, 5.99037067063811904637463777581, 6.90914754016952047170564110259, 7.27711862987395826528025364459