L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s − 6·6-s + 7·7-s + 8·8-s + 9·9-s + 28·11-s − 12·12-s + 86·13-s + 14·14-s + 16·16-s + 66·17-s + 18·18-s − 48·19-s − 21·21-s + 56·22-s − 140·23-s − 24·24-s + 172·26-s − 27·27-s + 28·28-s − 34·29-s − 284·31-s + 32·32-s − 84·33-s + 132·34-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 + 0.767·11-s − 0.288·12-s + 1.83·13-s + 0.267·14-s + 1/4·16-s + 0.941·17-s + 0.235·18-s − 0.579·19-s − 0.218·21-s + 0.542·22-s − 1.26·23-s − 0.204·24-s + 1.29·26-s − 0.192·27-s + 0.188·28-s − 0.217·29-s − 1.64·31-s + 0.176·32-s − 0.443·33-s + 0.665·34-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
\(L(2)\) |
\(\approx\) |
\(3.418616770\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.418616770\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 - 28 T + p^{3} T^{2} \) |
| 13 | \( 1 - 86 T + p^{3} T^{2} \) |
| 17 | \( 1 - 66 T + p^{3} T^{2} \) |
| 19 | \( 1 + 48 T + p^{3} T^{2} \) |
| 23 | \( 1 + 140 T + p^{3} T^{2} \) |
| 29 | \( 1 + 34 T + p^{3} T^{2} \) |
| 31 | \( 1 + 284 T + p^{3} T^{2} \) |
| 37 | \( 1 - 346 T + p^{3} T^{2} \) |
| 41 | \( 1 + 274 T + p^{3} T^{2} \) |
| 43 | \( 1 - 4 T + p^{3} T^{2} \) |
| 47 | \( 1 - 448 T + p^{3} T^{2} \) |
| 53 | \( 1 - 94 T + p^{3} T^{2} \) |
| 59 | \( 1 - 308 T + p^{3} T^{2} \) |
| 61 | \( 1 - 510 T + p^{3} T^{2} \) |
| 67 | \( 1 - 156 T + p^{3} T^{2} \) |
| 71 | \( 1 - 336 T + p^{3} T^{2} \) |
| 73 | \( 1 - 1170 T + p^{3} T^{2} \) |
| 79 | \( 1 - 16 T + p^{3} T^{2} \) |
| 83 | \( 1 + 772 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1630 T + p^{3} T^{2} \) |
| 97 | \( 1 + 110 T + p^{3} T^{2} \) |
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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
−9.652439395970681191975563744767, −8.607208483737477087044636024104, −7.79197684079408657981087881707, −6.71971758457290754906601073718, −5.98887589602538261878889158527, −5.39132331977696056066584175261, −4.06912125586414195891354822786, −3.66741942185649126825730288876, −1.98413071173727524494695816194, −0.965979689230596066880180188151,
0.965979689230596066880180188151, 1.98413071173727524494695816194, 3.66741942185649126825730288876, 4.06912125586414195891354822786, 5.39132331977696056066584175261, 5.98887589602538261878889158527, 6.71971758457290754906601073718, 7.79197684079408657981087881707, 8.607208483737477087044636024104, 9.652439395970681191975563744767