L(s) = 1 | + 3-s − 2·5-s + 9-s − 2·11-s − 2·13-s − 2·15-s + 6·17-s + 19-s + 2·23-s − 25-s + 27-s − 4·29-s − 8·31-s − 2·33-s + 2·37-s − 2·39-s − 8·41-s + 8·43-s − 2·45-s + 2·47-s − 7·49-s + 6·51-s + 4·53-s + 4·55-s + 57-s − 2·61-s + 4·65-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.894·5-s + 1/3·9-s − 0.603·11-s − 0.554·13-s − 0.516·15-s + 1.45·17-s + 0.229·19-s + 0.417·23-s − 1/5·25-s + 0.192·27-s − 0.742·29-s − 1.43·31-s − 0.348·33-s + 0.328·37-s − 0.320·39-s − 1.24·41-s + 1.21·43-s − 0.298·45-s + 0.291·47-s − 49-s + 0.840·51-s + 0.549·53-s + 0.539·55-s + 0.132·57-s − 0.256·61-s + 0.496·65-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 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−7.955564560013018311501607982325, −7.59630148696104627366241749629, −7.04261392429750856729502470223, −5.77544007688471730889081499385, −5.14485985091361969167190938423, −4.14408491099634338578038601587, −3.44208508440745725728555668749, −2.69718462256202326350630943118, −1.48208129217347642411083990257, 0,
1.48208129217347642411083990257, 2.69718462256202326350630943118, 3.44208508440745725728555668749, 4.14408491099634338578038601587, 5.14485985091361969167190938423, 5.77544007688471730889081499385, 7.04261392429750856729502470223, 7.59630148696104627366241749629, 7.955564560013018311501607982325