| L(s) = 1 | − 2-s + 2·3-s − 7·4-s − 16·5-s − 2·6-s + 7·7-s + 15·8-s − 23·9-s + 16·10-s + 8·11-s − 14·12-s + 28·13-s − 7·14-s − 32·15-s + 41·16-s + 23·18-s − 110·19-s + 112·20-s + 14·21-s − 8·22-s − 48·23-s + 30·24-s + 131·25-s − 28·26-s − 100·27-s − 49·28-s + 110·29-s + ⋯ |
| L(s) = 1 | − 0.353·2-s + 0.384·3-s − 7/8·4-s − 1.43·5-s − 0.136·6-s + 0.377·7-s + 0.662·8-s − 0.851·9-s + 0.505·10-s + 0.219·11-s − 0.336·12-s + 0.597·13-s − 0.133·14-s − 0.550·15-s + 0.640·16-s + 0.301·18-s − 1.32·19-s + 1.25·20-s + 0.145·21-s − 0.0775·22-s − 0.435·23-s + 0.255·24-s + 1.04·25-s − 0.211·26-s − 0.712·27-s − 0.330·28-s + 0.704·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2023 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 - p T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + T + p^{3} T^{2} \) |
| 3 | \( 1 - 2 T + p^{3} T^{2} \) |
| 5 | \( 1 + 16 T + p^{3} T^{2} \) |
| 11 | \( 1 - 8 T + p^{3} T^{2} \) |
| 13 | \( 1 - 28 T + p^{3} T^{2} \) |
| 19 | \( 1 + 110 T + p^{3} T^{2} \) |
| 23 | \( 1 + 48 T + p^{3} T^{2} \) |
| 29 | \( 1 - 110 T + p^{3} T^{2} \) |
| 31 | \( 1 + 12 T + p^{3} T^{2} \) |
| 37 | \( 1 - 246 T + p^{3} T^{2} \) |
| 41 | \( 1 + 182 T + p^{3} T^{2} \) |
| 43 | \( 1 - 128 T + p^{3} T^{2} \) |
| 47 | \( 1 - 324 T + p^{3} T^{2} \) |
| 53 | \( 1 + 162 T + p^{3} T^{2} \) |
| 59 | \( 1 - 810 T + p^{3} T^{2} \) |
| 61 | \( 1 - 8 p T + p^{3} T^{2} \) |
| 67 | \( 1 - 244 T + p^{3} T^{2} \) |
| 71 | \( 1 - 768 T + p^{3} T^{2} \) |
| 73 | \( 1 - 702 T + p^{3} T^{2} \) |
| 79 | \( 1 + 440 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1302 T + p^{3} T^{2} \) |
| 89 | \( 1 - 730 T + p^{3} T^{2} \) |
| 97 | \( 1 + 294 T + p^{3} T^{2} \) |
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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
−8.281089090805232835847017922864, −8.119469996871084004080088466547, −7.09348779906070825792719433088, −6.01567260665594557998558968020, −4.97653078235414337154472409882, −4.06109927005786865888711679968, −3.70045820096996972130890872822, −2.40758285035892424150129739254, −0.917578334401803458965068675174, 0,
0.917578334401803458965068675174, 2.40758285035892424150129739254, 3.70045820096996972130890872822, 4.06109927005786865888711679968, 4.97653078235414337154472409882, 6.01567260665594557998558968020, 7.09348779906070825792719433088, 8.119469996871084004080088466547, 8.281089090805232835847017922864
