| L(s) = 1 | − 22·7-s + 14·11-s + 30·13-s + 62·17-s − 120·19-s + 188·23-s − 96·29-s + 184·31-s − 406·37-s − 130·41-s − 148·43-s + 448·47-s + 141·49-s − 414·53-s − 266·59-s − 838·61-s − 248·67-s − 1.02e3·71-s − 484·73-s − 308·77-s − 48·79-s + 548·83-s + 650·89-s − 660·91-s + 1.81e3·97-s − 1.68e3·101-s + 298·103-s + ⋯ |
| L(s) = 1 | − 1.18·7-s + 0.383·11-s + 0.640·13-s + 0.884·17-s − 1.44·19-s + 1.70·23-s − 0.614·29-s + 1.06·31-s − 1.80·37-s − 0.495·41-s − 0.524·43-s + 1.39·47-s + 0.411·49-s − 1.07·53-s − 0.586·59-s − 1.75·61-s − 0.452·67-s − 1.70·71-s − 0.775·73-s − 0.455·77-s − 0.0683·79-s + 0.724·83-s + 0.774·89-s − 0.760·91-s + 1.90·97-s − 1.66·101-s + 0.285·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 22 T + p^{3} T^{2} \) |
| 11 | \( 1 - 14 T + p^{3} T^{2} \) |
| 13 | \( 1 - 30 T + p^{3} T^{2} \) |
| 17 | \( 1 - 62 T + p^{3} T^{2} \) |
| 19 | \( 1 + 120 T + p^{3} T^{2} \) |
| 23 | \( 1 - 188 T + p^{3} T^{2} \) |
| 29 | \( 1 + 96 T + p^{3} T^{2} \) |
| 31 | \( 1 - 184 T + p^{3} T^{2} \) |
| 37 | \( 1 + 406 T + p^{3} T^{2} \) |
| 41 | \( 1 + 130 T + p^{3} T^{2} \) |
| 43 | \( 1 + 148 T + p^{3} T^{2} \) |
| 47 | \( 1 - 448 T + p^{3} T^{2} \) |
| 53 | \( 1 + 414 T + p^{3} T^{2} \) |
| 59 | \( 1 + 266 T + p^{3} T^{2} \) |
| 61 | \( 1 + 838 T + p^{3} T^{2} \) |
| 67 | \( 1 + 248 T + p^{3} T^{2} \) |
| 71 | \( 1 + 1020 T + p^{3} T^{2} \) |
| 73 | \( 1 + 484 T + p^{3} T^{2} \) |
| 79 | \( 1 + 48 T + p^{3} T^{2} \) |
| 83 | \( 1 - 548 T + p^{3} T^{2} \) |
| 89 | \( 1 - 650 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1816 T + p^{3} T^{2} \) |
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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
−9.161785831462419252722957916942, −8.719532153645374580350524102807, −7.51010471249783789909434307221, −6.60064635526440518878699333779, −6.03217374992231065185497025269, −4.84010864228896918404101802234, −3.67358707023879935614062915546, −2.93227359806216301692346059997, −1.40524800629792220099746244878, 0,
1.40524800629792220099746244878, 2.93227359806216301692346059997, 3.67358707023879935614062915546, 4.84010864228896918404101802234, 6.03217374992231065185497025269, 6.60064635526440518878699333779, 7.51010471249783789909434307221, 8.719532153645374580350524102807, 9.161785831462419252722957916942
