| L(s) = 1 | + 2·2-s + 6·3-s + 4·4-s + 5·5-s + 12·6-s + 3·7-s + 8·8-s + 9·9-s + 10·10-s + 5·11-s + 24·12-s − 16·13-s + 6·14-s + 30·15-s + 16·16-s + 115·17-s + 18·18-s + 110·19-s + 20·20-s + 18·21-s + 10·22-s + 6·23-s + 48·24-s + 25·25-s − 32·26-s − 108·27-s + 12·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.447·5-s + 0.816·6-s + 0.161·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.137·11-s + 0.577·12-s − 0.341·13-s + 0.114·14-s + 0.516·15-s + 1/4·16-s + 1.64·17-s + 0.235·18-s + 1.32·19-s + 0.223·20-s + 0.187·21-s + 0.0969·22-s + 0.0543·23-s + 0.408·24-s + 1/5·25-s − 0.241·26-s − 0.769·27-s + 0.0809·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.828570616\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.828570616\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 5 | \( 1 - p T \) |
| 37 | \( 1 + p T \) |
| good | 3 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 7 | \( 1 - 3 T + p^{3} T^{2} \) |
| 11 | \( 1 - 5 T + p^{3} T^{2} \) |
| 13 | \( 1 + 16 T + p^{3} T^{2} \) |
| 17 | \( 1 - 115 T + p^{3} T^{2} \) |
| 19 | \( 1 - 110 T + p^{3} T^{2} \) |
| 23 | \( 1 - 6 T + p^{3} T^{2} \) |
| 29 | \( 1 + 111 T + p^{3} T^{2} \) |
| 31 | \( 1 + 79 T + p^{3} T^{2} \) |
| 41 | \( 1 - 171 T + p^{3} T^{2} \) |
| 43 | \( 1 - 361 T + p^{3} T^{2} \) |
| 47 | \( 1 + 428 T + p^{3} T^{2} \) |
| 53 | \( 1 + 527 T + p^{3} T^{2} \) |
| 59 | \( 1 - 112 T + p^{3} T^{2} \) |
| 61 | \( 1 + 323 T + p^{3} T^{2} \) |
| 67 | \( 1 + 464 T + p^{3} T^{2} \) |
| 71 | \( 1 + 366 T + p^{3} T^{2} \) |
| 73 | \( 1 - 712 T + p^{3} T^{2} \) |
| 79 | \( 1 - 176 T + p^{3} T^{2} \) |
| 83 | \( 1 + 180 T + p^{3} T^{2} \) |
| 89 | \( 1 - 446 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1407 T + p^{3} T^{2} \) |
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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
−11.07698268671899580953166161841, −9.826265303170820195509495976830, −9.275968695325707202942322714742, −7.966749450611971722521767209312, −7.40346452022804064960712416767, −5.96240520449186523617405982340, −5.06412854095817890440976791862, −3.60739024343069900028573732636, −2.84819511679162299229342548183, −1.52924601567157481188799166972,
1.52924601567157481188799166972, 2.84819511679162299229342548183, 3.60739024343069900028573732636, 5.06412854095817890440976791862, 5.96240520449186523617405982340, 7.40346452022804064960712416767, 7.966749450611971722521767209312, 9.275968695325707202942322714742, 9.826265303170820195509495976830, 11.07698268671899580953166161841
