L(s) = 1 | + 7-s + 4·11-s − 2·13-s − 4·17-s − 4·23-s − 5·25-s − 4·29-s − 8·31-s + 2·37-s + 4·41-s − 8·43-s − 8·47-s + 49-s + 4·53-s + 8·59-s + 2·61-s − 8·67-s − 12·71-s + 6·73-s + 4·77-s − 8·79-s + 16·83-s + 12·89-s − 2·91-s − 2·97-s − 8·103-s + 4·107-s + ⋯ |
L(s) = 1 | + 0.377·7-s + 1.20·11-s − 0.554·13-s − 0.970·17-s − 0.834·23-s − 25-s − 0.742·29-s − 1.43·31-s + 0.328·37-s + 0.624·41-s − 1.21·43-s − 1.16·47-s + 1/7·49-s + 0.549·53-s + 1.04·59-s + 0.256·61-s − 0.977·67-s − 1.42·71-s + 0.702·73-s + 0.455·77-s − 0.900·79-s + 1.75·83-s + 1.27·89-s − 0.209·91-s − 0.203·97-s − 0.788·103-s + 0.386·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 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
−8.040633251925508340780279940535, −7.35752941532385774081120297204, −6.61812151994534383055923893777, −5.90902380787712789094736520998, −5.05245576424501022426376997297, −4.16612973020341387783386317061, −3.61412835521968463843839470804, −2.28049444281587638337613384262, −1.57268571210906626664462069207, 0,
1.57268571210906626664462069207, 2.28049444281587638337613384262, 3.61412835521968463843839470804, 4.16612973020341387783386317061, 5.05245576424501022426376997297, 5.90902380787712789094736520998, 6.61812151994534383055923893777, 7.35752941532385774081120297204, 8.040633251925508340780279940535