| L(s) = 1 | − 2·2-s + 3·3-s + 4·4-s + 8·5-s − 6·6-s − 8·8-s + 9·9-s − 16·10-s + 40·11-s + 12·12-s + 4·13-s + 24·15-s + 16·16-s − 84·17-s − 18·18-s + 148·19-s + 32·20-s − 80·22-s + 84·23-s − 24·24-s − 61·25-s − 8·26-s + 27·27-s + 58·29-s − 48·30-s − 136·31-s − 32·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.715·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.505·10-s + 1.09·11-s + 0.288·12-s + 0.0853·13-s + 0.413·15-s + 1/4·16-s − 1.19·17-s − 0.235·18-s + 1.78·19-s + 0.357·20-s − 0.775·22-s + 0.761·23-s − 0.204·24-s − 0.487·25-s − 0.0603·26-s + 0.192·27-s + 0.371·29-s − 0.292·30-s − 0.787·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.006687205\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.006687205\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 3 | \( 1 - p T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 8 T + p^{3} T^{2} \) |
| 11 | \( 1 - 40 T + p^{3} T^{2} \) |
| 13 | \( 1 - 4 T + p^{3} T^{2} \) |
| 17 | \( 1 + 84 T + p^{3} T^{2} \) |
| 19 | \( 1 - 148 T + p^{3} T^{2} \) |
| 23 | \( 1 - 84 T + p^{3} T^{2} \) |
| 29 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 31 | \( 1 + 136 T + p^{3} T^{2} \) |
| 37 | \( 1 + 6 p T + p^{3} T^{2} \) |
| 41 | \( 1 - 420 T + p^{3} T^{2} \) |
| 43 | \( 1 + 164 T + p^{3} T^{2} \) |
| 47 | \( 1 - 488 T + p^{3} T^{2} \) |
| 53 | \( 1 - 478 T + p^{3} T^{2} \) |
| 59 | \( 1 - 548 T + p^{3} T^{2} \) |
| 61 | \( 1 - 692 T + p^{3} T^{2} \) |
| 67 | \( 1 + 908 T + p^{3} T^{2} \) |
| 71 | \( 1 + 524 T + p^{3} T^{2} \) |
| 73 | \( 1 - 440 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1216 T + p^{3} T^{2} \) |
| 83 | \( 1 + 684 T + p^{3} T^{2} \) |
| 89 | \( 1 - 604 T + p^{3} T^{2} \) |
| 97 | \( 1 + 832 T + p^{3} T^{2} \) |
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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
−11.22894754926272745903460195339, −10.15224328898191139434742000561, −9.241355611549007854133376466335, −8.855951246216007523497240138527, −7.48147140812442966306247349246, −6.67955519858019177128485458236, −5.46100288531879534106624052515, −3.83953552024101295380785475129, −2.43457001579993789323890589457, −1.17754236925011366554141561200,
1.17754236925011366554141561200, 2.43457001579993789323890589457, 3.83953552024101295380785475129, 5.46100288531879534106624052515, 6.67955519858019177128485458236, 7.48147140812442966306247349246, 8.855951246216007523497240138527, 9.241355611549007854133376466335, 10.15224328898191139434742000561, 11.22894754926272745903460195339
