L(s) = 1 | + 4·3-s − 2·5-s − 24·7-s − 11·9-s + 44·11-s + 22·13-s − 8·15-s + 50·17-s − 44·19-s − 96·21-s + 56·23-s − 121·25-s − 152·27-s + 198·29-s + 160·31-s + 176·33-s + 48·35-s − 162·37-s + 88·39-s − 198·41-s − 52·43-s + 22·45-s − 528·47-s + 233·49-s + 200·51-s − 242·53-s − 88·55-s + ⋯ |
L(s) = 1 | + 0.769·3-s − 0.178·5-s − 1.29·7-s − 0.407·9-s + 1.20·11-s + 0.469·13-s − 0.137·15-s + 0.713·17-s − 0.531·19-s − 0.997·21-s + 0.507·23-s − 0.967·25-s − 1.08·27-s + 1.26·29-s + 0.926·31-s + 0.928·33-s + 0.231·35-s − 0.719·37-s + 0.361·39-s − 0.754·41-s − 0.184·43-s + 0.0728·45-s − 1.63·47-s + 0.679·49-s + 0.549·51-s − 0.627·53-s − 0.215·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.113696743\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.113696743\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 - 4 T + p^{3} T^{2} \) |
| 5 | \( 1 + 2 T + p^{3} T^{2} \) |
| 7 | \( 1 + 24 T + p^{3} T^{2} \) |
| 11 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 13 | \( 1 - 22 T + p^{3} T^{2} \) |
| 17 | \( 1 - 50 T + p^{3} T^{2} \) |
| 19 | \( 1 + 44 T + p^{3} T^{2} \) |
| 23 | \( 1 - 56 T + p^{3} T^{2} \) |
| 29 | \( 1 - 198 T + p^{3} T^{2} \) |
| 31 | \( 1 - 160 T + p^{3} T^{2} \) |
| 37 | \( 1 + 162 T + p^{3} T^{2} \) |
| 41 | \( 1 + 198 T + p^{3} T^{2} \) |
| 43 | \( 1 + 52 T + p^{3} T^{2} \) |
| 47 | \( 1 + 528 T + p^{3} T^{2} \) |
| 53 | \( 1 + 242 T + p^{3} T^{2} \) |
| 59 | \( 1 - 668 T + p^{3} T^{2} \) |
| 61 | \( 1 - 550 T + p^{3} T^{2} \) |
| 67 | \( 1 + 188 T + p^{3} T^{2} \) |
| 71 | \( 1 + 728 T + p^{3} T^{2} \) |
| 73 | \( 1 - 154 T + p^{3} T^{2} \) |
| 79 | \( 1 - 656 T + p^{3} T^{2} \) |
| 83 | \( 1 + 236 T + p^{3} T^{2} \) |
| 89 | \( 1 - 714 T + p^{3} T^{2} \) |
| 97 | \( 1 + 478 T + p^{3} T^{2} \) |
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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
−19.22215472794746889200943014736, −17.25565393065787159096779088380, −15.97767826239660211610183387568, −14.59342058368142683805922761806, −13.39677486933593777643407800038, −11.85997478385713065617074576105, −9.821732387528819879529923492338, −8.516504256180230350948942627864, −6.46458853043492391741791065933, −3.43045078283104518723714384423,
3.43045078283104518723714384423, 6.46458853043492391741791065933, 8.516504256180230350948942627864, 9.821732387528819879529923492338, 11.85997478385713065617074576105, 13.39677486933593777643407800038, 14.59342058368142683805922761806, 15.97767826239660211610183387568, 17.25565393065787159096779088380, 19.22215472794746889200943014736