| L(s) = 1 | − 4·5-s + 7·7-s − 12·11-s + 27·13-s + 61·17-s − 62·19-s − 13·23-s − 109·25-s − 19·29-s − 329·31-s − 28·35-s + 42·37-s + 310·41-s + 491·43-s + 290·47-s + 49·49-s − 373·53-s + 48·55-s + 111·59-s − 254·61-s − 108·65-s − 297·67-s + 93·71-s − 254·73-s − 84·77-s − 614·79-s − 1.38e3·83-s + ⋯ |
| L(s) = 1 | − 0.357·5-s + 0.377·7-s − 0.328·11-s + 0.576·13-s + 0.870·17-s − 0.748·19-s − 0.117·23-s − 0.871·25-s − 0.121·29-s − 1.90·31-s − 0.135·35-s + 0.186·37-s + 1.18·41-s + 1.74·43-s + 0.900·47-s + 1/7·49-s − 0.966·53-s + 0.117·55-s + 0.244·59-s − 0.533·61-s − 0.206·65-s − 0.541·67-s + 0.155·71-s − 0.407·73-s − 0.124·77-s − 0.874·79-s − 1.82·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - p T \) |
| good | 5 | \( 1 + 4 T + p^{3} T^{2} \) |
| 11 | \( 1 + 12 T + p^{3} T^{2} \) |
| 13 | \( 1 - 27 T + p^{3} T^{2} \) |
| 17 | \( 1 - 61 T + p^{3} T^{2} \) |
| 19 | \( 1 + 62 T + p^{3} T^{2} \) |
| 23 | \( 1 + 13 T + p^{3} T^{2} \) |
| 29 | \( 1 + 19 T + p^{3} T^{2} \) |
| 31 | \( 1 + 329 T + p^{3} T^{2} \) |
| 37 | \( 1 - 42 T + p^{3} T^{2} \) |
| 41 | \( 1 - 310 T + p^{3} T^{2} \) |
| 43 | \( 1 - 491 T + p^{3} T^{2} \) |
| 47 | \( 1 - 290 T + p^{3} T^{2} \) |
| 53 | \( 1 + 373 T + p^{3} T^{2} \) |
| 59 | \( 1 - 111 T + p^{3} T^{2} \) |
| 61 | \( 1 + 254 T + p^{3} T^{2} \) |
| 67 | \( 1 + 297 T + p^{3} T^{2} \) |
| 71 | \( 1 - 93 T + p^{3} T^{2} \) |
| 73 | \( 1 + 254 T + p^{3} T^{2} \) |
| 79 | \( 1 + 614 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1380 T + p^{3} T^{2} \) |
| 89 | \( 1 - T + p^{3} T^{2} \) |
| 97 | \( 1 + 488 T + p^{3} T^{2} \) |
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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
−8.716302662538766670762412431908, −7.75671496826380939474770842200, −7.38897701911000281200115001990, −6.07695144800745119247101561474, −5.53167245173770478134265532752, −4.34785541031158847846089698897, −3.66652301188144604024415600409, −2.46390892828116164518446291193, −1.32606508077459305315742886643, 0,
1.32606508077459305315742886643, 2.46390892828116164518446291193, 3.66652301188144604024415600409, 4.34785541031158847846089698897, 5.53167245173770478134265532752, 6.07695144800745119247101561474, 7.38897701911000281200115001990, 7.75671496826380939474770842200, 8.716302662538766670762412431908
