| L(s) = 1 | − 3·3-s − 10·7-s + 9·9-s + 46·11-s − 34·13-s + 66·17-s − 104·19-s + 30·21-s − 164·23-s − 27·27-s + 224·29-s + 72·31-s − 138·33-s − 22·37-s + 102·39-s + 194·41-s − 108·43-s + 480·47-s − 243·49-s − 198·51-s + 286·53-s + 312·57-s − 426·59-s + 698·61-s − 90·63-s − 328·67-s + 492·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.539·7-s + 1/3·9-s + 1.26·11-s − 0.725·13-s + 0.941·17-s − 1.25·19-s + 0.311·21-s − 1.48·23-s − 0.192·27-s + 1.43·29-s + 0.417·31-s − 0.727·33-s − 0.0977·37-s + 0.418·39-s + 0.738·41-s − 0.383·43-s + 1.48·47-s − 0.708·49-s − 0.543·51-s + 0.741·53-s + 0.725·57-s − 0.940·59-s + 1.46·61-s − 0.179·63-s − 0.598·67-s + 0.858·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 10 T + p^{3} T^{2} \) |
| 11 | \( 1 - 46 T + p^{3} T^{2} \) |
| 13 | \( 1 + 34 T + p^{3} T^{2} \) |
| 17 | \( 1 - 66 T + p^{3} T^{2} \) |
| 19 | \( 1 + 104 T + p^{3} T^{2} \) |
| 23 | \( 1 + 164 T + p^{3} T^{2} \) |
| 29 | \( 1 - 224 T + p^{3} T^{2} \) |
| 31 | \( 1 - 72 T + p^{3} T^{2} \) |
| 37 | \( 1 + 22 T + p^{3} T^{2} \) |
| 41 | \( 1 - 194 T + p^{3} T^{2} \) |
| 43 | \( 1 + 108 T + p^{3} T^{2} \) |
| 47 | \( 1 - 480 T + p^{3} T^{2} \) |
| 53 | \( 1 - 286 T + p^{3} T^{2} \) |
| 59 | \( 1 + 426 T + p^{3} T^{2} \) |
| 61 | \( 1 - 698 T + p^{3} T^{2} \) |
| 67 | \( 1 + 328 T + p^{3} T^{2} \) |
| 71 | \( 1 + 188 T + p^{3} T^{2} \) |
| 73 | \( 1 + 740 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1168 T + p^{3} T^{2} \) |
| 83 | \( 1 + 412 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1206 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1384 T + p^{3} T^{2} \) |
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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
−9.041056047962812988723373276776, −8.158742672835397609184854016065, −7.13797839556417246695348517062, −6.37762354123565371483459210891, −5.78360378119950164451846780587, −4.54211796593548807409055654972, −3.85791401515934591379026062499, −2.55904891088663822032084220208, −1.25302284592991174915457073030, 0,
1.25302284592991174915457073030, 2.55904891088663822032084220208, 3.85791401515934591379026062499, 4.54211796593548807409055654972, 5.78360378119950164451846780587, 6.37762354123565371483459210891, 7.13797839556417246695348517062, 8.158742672835397609184854016065, 9.041056047962812988723373276776
