| L(s) = 1 | − 2·5-s − 24·7-s + 44·11-s − 22·13-s − 50·17-s + 44·19-s − 56·23-s − 121·25-s + 198·29-s + 160·31-s + 48·35-s + 162·37-s + 198·41-s + 52·43-s + 528·47-s + 233·49-s − 242·53-s − 88·55-s + 668·59-s − 550·61-s + 44·65-s + 188·67-s + 728·71-s + 154·73-s − 1.05e3·77-s + 656·79-s − 236·83-s + ⋯ |
| L(s) = 1 | − 0.178·5-s − 1.29·7-s + 1.20·11-s − 0.469·13-s − 0.713·17-s + 0.531·19-s − 0.507·23-s − 0.967·25-s + 1.26·29-s + 0.926·31-s + 0.231·35-s + 0.719·37-s + 0.754·41-s + 0.184·43-s + 1.63·47-s + 0.679·49-s − 0.627·53-s − 0.215·55-s + 1.47·59-s − 1.15·61-s + 0.0839·65-s + 0.342·67-s + 1.21·71-s + 0.246·73-s − 1.56·77-s + 0.934·79-s − 0.312·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.502421561\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.502421561\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 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
−10.07098916610288391649839395089, −9.552194496875298998545375136560, −8.706813470069115477357016290695, −7.54524289754732726513256712802, −6.59057568545276685794570228985, −6.00476622722264718226748341380, −4.50914052460823240544076136086, −3.61364281072863053067279191630, −2.44662097748618588863201330406, −0.72792429959464475730524534411,
0.72792429959464475730524534411, 2.44662097748618588863201330406, 3.61364281072863053067279191630, 4.50914052460823240544076136086, 6.00476622722264718226748341380, 6.59057568545276685794570228985, 7.54524289754732726513256712802, 8.706813470069115477357016290695, 9.552194496875298998545375136560, 10.07098916610288391649839395089
