L(s) = 1 | − 2·2-s + 4·4-s + 4·7-s − 8·8-s + 42·11-s − 20·13-s − 8·14-s + 16·16-s − 93·17-s + 59·19-s − 84·22-s − 9·23-s + 40·26-s + 16·28-s + 120·29-s + 47·31-s − 32·32-s + 186·34-s + 262·37-s − 118·38-s + 126·41-s + 178·43-s + 168·44-s + 18·46-s − 144·47-s − 327·49-s − 80·52-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.215·7-s − 0.353·8-s + 1.15·11-s − 0.426·13-s − 0.152·14-s + 1/4·16-s − 1.32·17-s + 0.712·19-s − 0.814·22-s − 0.0815·23-s + 0.301·26-s + 0.107·28-s + 0.768·29-s + 0.272·31-s − 0.176·32-s + 0.938·34-s + 1.16·37-s − 0.503·38-s + 0.479·41-s + 0.631·43-s + 0.575·44-s + 0.0576·46-s − 0.446·47-s − 0.953·49-s − 0.213·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.538904932\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.538904932\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 4 T + p^{3} T^{2} \) |
| 11 | \( 1 - 42 T + p^{3} T^{2} \) |
| 13 | \( 1 + 20 T + p^{3} T^{2} \) |
| 17 | \( 1 + 93 T + p^{3} T^{2} \) |
| 19 | \( 1 - 59 T + p^{3} T^{2} \) |
| 23 | \( 1 + 9 T + p^{3} T^{2} \) |
| 29 | \( 1 - 120 T + p^{3} T^{2} \) |
| 31 | \( 1 - 47 T + p^{3} T^{2} \) |
| 37 | \( 1 - 262 T + p^{3} T^{2} \) |
| 41 | \( 1 - 126 T + p^{3} T^{2} \) |
| 43 | \( 1 - 178 T + p^{3} T^{2} \) |
| 47 | \( 1 + 144 T + p^{3} T^{2} \) |
| 53 | \( 1 + 741 T + p^{3} T^{2} \) |
| 59 | \( 1 + 444 T + p^{3} T^{2} \) |
| 61 | \( 1 - 221 T + p^{3} T^{2} \) |
| 67 | \( 1 - 538 T + p^{3} T^{2} \) |
| 71 | \( 1 - 690 T + p^{3} T^{2} \) |
| 73 | \( 1 - 1126 T + p^{3} T^{2} \) |
| 79 | \( 1 - 665 T + p^{3} T^{2} \) |
| 83 | \( 1 + 75 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1086 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1544 T + p^{3} T^{2} \) |
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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
−9.467835461633746621424468358665, −8.437993202191697606375138507315, −7.76523644792374253933170044545, −6.74411651543278410681089900793, −6.28149279374059948248823924114, −4.98710522662398708720151522974, −4.09589744469321353775115573569, −2.87829359401321948539198734684, −1.79887497039630474965602554863, −0.70267969175486500416557330567,
0.70267969175486500416557330567, 1.79887497039630474965602554863, 2.87829359401321948539198734684, 4.09589744469321353775115573569, 4.98710522662398708720151522974, 6.28149279374059948248823924114, 6.74411651543278410681089900793, 7.76523644792374253933170044545, 8.437993202191697606375138507315, 9.467835461633746621424468358665