| L(s) = 1 | + 3·3-s − 5·5-s − 8.87·7-s + 9·9-s − 31.0·11-s + 8.98·13-s − 15·15-s + 45.3·17-s + 46.4·19-s − 26.6·21-s + 23·23-s + 25·25-s + 27·27-s + 184.·29-s − 31.1·31-s − 93.1·33-s + 44.3·35-s − 230.·37-s + 26.9·39-s − 145.·41-s + 445.·43-s − 45·45-s − 450.·47-s − 264.·49-s + 136.·51-s + 45.3·53-s + 155.·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.479·7-s + 0.333·9-s − 0.850·11-s + 0.191·13-s − 0.258·15-s + 0.647·17-s + 0.560·19-s − 0.276·21-s + 0.208·23-s + 0.200·25-s + 0.192·27-s + 1.17·29-s − 0.180·31-s − 0.491·33-s + 0.214·35-s − 1.02·37-s + 0.110·39-s − 0.555·41-s + 1.57·43-s − 0.149·45-s − 1.39·47-s − 0.770·49-s + 0.373·51-s + 0.117·53-s + 0.380·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 3T \) |
| 5 | \( 1 + 5T \) |
| 23 | \( 1 - 23T \) |
| good | 7 | \( 1 + 8.87T + 343T^{2} \) |
| 11 | \( 1 + 31.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 8.98T + 2.19e3T^{2} \) |
| 17 | \( 1 - 45.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 46.4T + 6.85e3T^{2} \) |
| 29 | \( 1 - 184.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 31.1T + 2.97e4T^{2} \) |
| 37 | \( 1 + 230.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 145.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 445.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 450.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 45.3T + 1.48e5T^{2} \) |
| 59 | \( 1 + 262.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 694.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 798.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 400.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 652.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 38.2T + 4.93e5T^{2} \) |
| 83 | \( 1 - 316.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 778.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.58e3T + 9.12e5T^{2} \) |
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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
−8.744303271803563674472443161703, −7.983112746587369726313307717459, −7.36278451747326128012518771263, −6.44025373425463254982577855734, −5.39596793073786343952149424181, −4.48642572882946448749173929088, −3.35964619954062763029043049024, −2.80275396464652530064440739473, −1.37333213063780606698227460867, 0,
1.37333213063780606698227460867, 2.80275396464652530064440739473, 3.35964619954062763029043049024, 4.48642572882946448749173929088, 5.39596793073786343952149424181, 6.44025373425463254982577855734, 7.36278451747326128012518771263, 7.983112746587369726313307717459, 8.744303271803563674472443161703
