| L(s) = 1 | + 2·3-s − 26·7-s − 23·9-s − 28·11-s − 12·13-s − 64·17-s − 60·19-s − 52·21-s + 58·23-s − 100·27-s − 90·29-s + 128·31-s − 56·33-s − 236·37-s − 24·39-s + 242·41-s + 362·43-s − 226·47-s + 333·49-s − 128·51-s + 108·53-s − 120·57-s − 20·59-s − 542·61-s + 598·63-s − 434·67-s + 116·69-s + ⋯ |
| L(s) = 1 | + 0.384·3-s − 1.40·7-s − 0.851·9-s − 0.767·11-s − 0.256·13-s − 0.913·17-s − 0.724·19-s − 0.540·21-s + 0.525·23-s − 0.712·27-s − 0.576·29-s + 0.741·31-s − 0.295·33-s − 1.04·37-s − 0.0985·39-s + 0.921·41-s + 1.28·43-s − 0.701·47-s + 0.970·49-s − 0.351·51-s + 0.279·53-s − 0.278·57-s − 0.0441·59-s − 1.13·61-s + 1.19·63-s − 0.791·67-s + 0.202·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8150365103\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8150365103\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p^{3} T^{2} \) |
| 7 | \( 1 + 26 T + p^{3} T^{2} \) |
| 11 | \( 1 + 28 T + p^{3} T^{2} \) |
| 13 | \( 1 + 12 T + p^{3} T^{2} \) |
| 17 | \( 1 + 64 T + p^{3} T^{2} \) |
| 19 | \( 1 + 60 T + p^{3} T^{2} \) |
| 23 | \( 1 - 58 T + p^{3} T^{2} \) |
| 29 | \( 1 + 90 T + p^{3} T^{2} \) |
| 31 | \( 1 - 128 T + p^{3} T^{2} \) |
| 37 | \( 1 + 236 T + p^{3} T^{2} \) |
| 41 | \( 1 - 242 T + p^{3} T^{2} \) |
| 43 | \( 1 - 362 T + p^{3} T^{2} \) |
| 47 | \( 1 + 226 T + p^{3} T^{2} \) |
| 53 | \( 1 - 108 T + p^{3} T^{2} \) |
| 59 | \( 1 + 20 T + p^{3} T^{2} \) |
| 61 | \( 1 + 542 T + p^{3} T^{2} \) |
| 67 | \( 1 + 434 T + p^{3} T^{2} \) |
| 71 | \( 1 - 1128 T + p^{3} T^{2} \) |
| 73 | \( 1 - 632 T + p^{3} T^{2} \) |
| 79 | \( 1 - 720 T + p^{3} T^{2} \) |
| 83 | \( 1 + 478 T + p^{3} T^{2} \) |
| 89 | \( 1 + 490 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1456 T + p^{3} T^{2} \) |
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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
−9.091664289665640065437549244409, −8.360645285530265661178043368799, −7.45957465379418827471691747938, −6.56565085334744627512392232436, −5.92635029375366982747429211298, −4.92510337604195680295580437135, −3.79249977571853771674370656849, −2.90879686690684928297338092565, −2.25606953382465632073228379385, −0.39426574137207148776708072076,
0.39426574137207148776708072076, 2.25606953382465632073228379385, 2.90879686690684928297338092565, 3.79249977571853771674370656849, 4.92510337604195680295580437135, 5.92635029375366982747429211298, 6.56565085334744627512392232436, 7.45957465379418827471691747938, 8.360645285530265661178043368799, 9.091664289665640065437549244409
