L(s) = 1 | + 3-s + 5-s + 9-s − 4·11-s − 7·13-s + 15-s + 6·17-s − 3·19-s − 2·23-s + 25-s + 27-s − 2·29-s − 7·31-s − 4·33-s − 7·37-s − 7·39-s + 8·41-s + 5·43-s + 45-s − 10·47-s + 6·51-s − 8·53-s − 4·55-s − 3·57-s − 10·59-s + 6·61-s − 7·65-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1/3·9-s − 1.20·11-s − 1.94·13-s + 0.258·15-s + 1.45·17-s − 0.688·19-s − 0.417·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s − 1.25·31-s − 0.696·33-s − 1.15·37-s − 1.12·39-s + 1.24·41-s + 0.762·43-s + 0.149·45-s − 1.45·47-s + 0.840·51-s − 1.09·53-s − 0.539·55-s − 0.397·57-s − 1.30·59-s + 0.768·61-s − 0.868·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 7 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 15 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 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
−8.264234219052835662840524102271, −7.58038405978316471743308736553, −7.19601002521936360224755429218, −5.94982713858785099491107604520, −5.25972398836122273924768521712, −4.56272009636257024202589978944, −3.35264561435067538664390565583, −2.57901384661548545444883920131, −1.78693567527845542392800100347, 0,
1.78693567527845542392800100347, 2.57901384661548545444883920131, 3.35264561435067538664390565583, 4.56272009636257024202589978944, 5.25972398836122273924768521712, 5.94982713858785099491107604520, 7.19601002521936360224755429218, 7.58038405978316471743308736553, 8.264234219052835662840524102271