| L(s) = 1 | − 2·2-s + 3·3-s + 4·4-s + 5·5-s − 6·6-s + 32·7-s − 8·8-s + 9·9-s − 10·10-s − 60·11-s + 12·12-s − 34·13-s − 64·14-s + 15·15-s + 16·16-s + 42·17-s − 18·18-s − 76·19-s + 20·20-s + 96·21-s + 120·22-s − 24·24-s + 25·25-s + 68·26-s + 27·27-s + 128·28-s + 6·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 1.72·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.64·11-s + 0.288·12-s − 0.725·13-s − 1.22·14-s + 0.258·15-s + 1/4·16-s + 0.599·17-s − 0.235·18-s − 0.917·19-s + 0.223·20-s + 0.997·21-s + 1.16·22-s − 0.204·24-s + 1/5·25-s + 0.512·26-s + 0.192·27-s + 0.863·28-s + 0.0384·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.164252130\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.164252130\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - p T \) |
| 5 | \( 1 - p T \) |
| good | 7 | \( 1 - 32 T + p^{3} T^{2} \) |
| 11 | \( 1 + 60 T + p^{3} T^{2} \) |
| 13 | \( 1 + 34 T + p^{3} T^{2} \) |
| 17 | \( 1 - 42 T + p^{3} T^{2} \) |
| 19 | \( 1 + 4 p T + p^{3} T^{2} \) |
| 23 | \( 1 + p^{3} T^{2} \) |
| 29 | \( 1 - 6 T + p^{3} T^{2} \) |
| 31 | \( 1 + 232 T + p^{3} T^{2} \) |
| 37 | \( 1 - 134 T + p^{3} T^{2} \) |
| 41 | \( 1 - 234 T + p^{3} T^{2} \) |
| 43 | \( 1 + 412 T + p^{3} T^{2} \) |
| 47 | \( 1 + 360 T + p^{3} T^{2} \) |
| 53 | \( 1 - 222 T + p^{3} T^{2} \) |
| 59 | \( 1 - 660 T + p^{3} T^{2} \) |
| 61 | \( 1 + 490 T + p^{3} T^{2} \) |
| 67 | \( 1 - 812 T + p^{3} T^{2} \) |
| 71 | \( 1 - 120 T + p^{3} T^{2} \) |
| 73 | \( 1 - 746 T + p^{3} T^{2} \) |
| 79 | \( 1 - 152 T + p^{3} T^{2} \) |
| 83 | \( 1 + 804 T + p^{3} T^{2} \) |
| 89 | \( 1 + 678 T + p^{3} T^{2} \) |
| 97 | \( 1 - 2 p T + p^{3} T^{2} \) |
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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
−16.73333217515122542071631352064, −15.18427938561334980621665534069, −14.36550246758532564408033643751, −12.83978191962661469621805694296, −11.15059243829229966840425042378, −10.04921028817379948976437709995, −8.429223868668413900752750168558, −7.57368581901634818634839656776, −5.15085829225291237322003695490, −2.16239651492029636231625026431,
2.16239651492029636231625026431, 5.15085829225291237322003695490, 7.57368581901634818634839656776, 8.429223868668413900752750168558, 10.04921028817379948976437709995, 11.15059243829229966840425042378, 12.83978191962661469621805694296, 14.36550246758532564408033643751, 15.18427938561334980621665534069, 16.73333217515122542071631352064
