| L(s) = 1 | + 2·2-s + 3·3-s + 4·4-s − 10·5-s + 6·6-s + 8·7-s + 8·8-s + 9·9-s − 20·10-s − 40·11-s + 12·12-s + 16·14-s − 30·15-s + 16·16-s + 130·17-s + 18·18-s + 20·19-s − 40·20-s + 24·21-s − 80·22-s + 24·24-s − 25·25-s + 27·27-s + 32·28-s − 18·29-s − 60·30-s + 184·31-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s + 0.431·7-s + 0.353·8-s + 1/3·9-s − 0.632·10-s − 1.09·11-s + 0.288·12-s + 0.305·14-s − 0.516·15-s + 1/4·16-s + 1.85·17-s + 0.235·18-s + 0.241·19-s − 0.447·20-s + 0.249·21-s − 0.775·22-s + 0.204·24-s − 1/5·25-s + 0.192·27-s + 0.215·28-s − 0.115·29-s − 0.365·30-s + 1.06·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.758857780\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.758857780\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - p T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 7 | \( 1 - 8 T + p^{3} T^{2} \) |
| 11 | \( 1 + 40 T + p^{3} T^{2} \) |
| 17 | \( 1 - 130 T + p^{3} T^{2} \) |
| 19 | \( 1 - 20 T + p^{3} T^{2} \) |
| 23 | \( 1 + p^{3} T^{2} \) |
| 29 | \( 1 + 18 T + p^{3} T^{2} \) |
| 31 | \( 1 - 184 T + p^{3} T^{2} \) |
| 37 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 41 | \( 1 - 362 T + p^{3} T^{2} \) |
| 43 | \( 1 - 76 T + p^{3} T^{2} \) |
| 47 | \( 1 - 452 T + p^{3} T^{2} \) |
| 53 | \( 1 - 382 T + p^{3} T^{2} \) |
| 59 | \( 1 + 464 T + p^{3} T^{2} \) |
| 61 | \( 1 - 358 T + p^{3} T^{2} \) |
| 67 | \( 1 - 700 T + p^{3} T^{2} \) |
| 71 | \( 1 - 748 T + p^{3} T^{2} \) |
| 73 | \( 1 + 1058 T + p^{3} T^{2} \) |
| 79 | \( 1 + 976 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1008 T + p^{3} T^{2} \) |
| 89 | \( 1 - 386 T + p^{3} T^{2} \) |
| 97 | \( 1 - 614 T + p^{3} T^{2} \) |
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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
−9.703974263994165601957947360196, −8.435113713446498097304087181019, −7.72330190374658220047622912186, −7.41812313424302815665877894826, −5.96954528153293445114661133586, −5.12485242198336402734946153796, −4.18184017096352409463594465195, −3.31826065918618045217441523449, −2.43039462383601003556197025285, −0.925280667783231477268077905844,
0.925280667783231477268077905844, 2.43039462383601003556197025285, 3.31826065918618045217441523449, 4.18184017096352409463594465195, 5.12485242198336402734946153796, 5.96954528153293445114661133586, 7.41812313424302815665877894826, 7.72330190374658220047622912186, 8.435113713446498097304087181019, 9.703974263994165601957947360196
