| L(s) = 1 | + 2·2-s + 3·3-s + 4·4-s + 6·6-s − 7·7-s + 8·8-s + 9·9-s + 42·11-s + 12·12-s − 47·13-s − 14·14-s + 16·16-s + 3·17-s + 18·18-s + 56·19-s − 21·21-s + 84·22-s − 9·23-s + 24·24-s − 94·26-s + 27·27-s − 28·28-s + 189·29-s + 263·31-s + 32·32-s + 126·33-s + 6·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 1.15·11-s + 0.288·12-s − 1.00·13-s − 0.267·14-s + 1/4·16-s + 0.0428·17-s + 0.235·18-s + 0.676·19-s − 0.218·21-s + 0.814·22-s − 0.0815·23-s + 0.204·24-s − 0.709·26-s + 0.192·27-s − 0.188·28-s + 1.21·29-s + 1.52·31-s + 0.176·32-s + 0.664·33-s + 0.0302·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.432174228\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.432174228\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - p T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + p T \) |
| good | 11 | \( 1 - 42 T + p^{3} T^{2} \) |
| 13 | \( 1 + 47 T + p^{3} T^{2} \) |
| 17 | \( 1 - 3 T + p^{3} T^{2} \) |
| 19 | \( 1 - 56 T + p^{3} T^{2} \) |
| 23 | \( 1 + 9 T + p^{3} T^{2} \) |
| 29 | \( 1 - 189 T + p^{3} T^{2} \) |
| 31 | \( 1 - 263 T + p^{3} T^{2} \) |
| 37 | \( 1 - 58 T + p^{3} T^{2} \) |
| 41 | \( 1 + 273 T + p^{3} T^{2} \) |
| 43 | \( 1 - 307 T + p^{3} T^{2} \) |
| 47 | \( 1 - 156 T + p^{3} T^{2} \) |
| 53 | \( 1 + 207 T + p^{3} T^{2} \) |
| 59 | \( 1 + 507 T + p^{3} T^{2} \) |
| 61 | \( 1 - 635 T + p^{3} T^{2} \) |
| 67 | \( 1 - 556 T + p^{3} T^{2} \) |
| 71 | \( 1 - 684 T + p^{3} T^{2} \) |
| 73 | \( 1 + 482 T + p^{3} T^{2} \) |
| 79 | \( 1 - 182 T + p^{3} T^{2} \) |
| 83 | \( 1 - 291 T + p^{3} T^{2} \) |
| 89 | \( 1 + 810 T + p^{3} T^{2} \) |
| 97 | \( 1 - 910 T + p^{3} T^{2} \) |
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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
−9.648743282996360547788992012694, −8.712982805976774136834196791114, −7.76428121964631386963415521127, −6.88853572994826142048877659003, −6.24290623814879315631344305515, −5.02983900895836521777083417405, −4.20977950581219246265782568707, −3.24369027118336069306940252427, −2.37456382838151215937828261360, −1.01831259290386358115806770016,
1.01831259290386358115806770016, 2.37456382838151215937828261360, 3.24369027118336069306940252427, 4.20977950581219246265782568707, 5.02983900895836521777083417405, 6.24290623814879315631344305515, 6.88853572994826142048877659003, 7.76428121964631386963415521127, 8.712982805976774136834196791114, 9.648743282996360547788992012694
