| L(s) = 1 | − 3·3-s − 8·7-s + 9·9-s + 20·11-s − 22·13-s + 14·17-s + 76·19-s + 24·21-s − 56·23-s − 27·27-s − 154·29-s + 160·31-s − 60·33-s + 162·37-s + 66·39-s − 390·41-s − 388·43-s + 544·47-s − 279·49-s − 42·51-s + 210·53-s − 228·57-s − 380·59-s − 794·61-s − 72·63-s + 148·67-s + 168·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.431·7-s + 1/3·9-s + 0.548·11-s − 0.469·13-s + 0.199·17-s + 0.917·19-s + 0.249·21-s − 0.507·23-s − 0.192·27-s − 0.986·29-s + 0.926·31-s − 0.316·33-s + 0.719·37-s + 0.270·39-s − 1.48·41-s − 1.37·43-s + 1.68·47-s − 0.813·49-s − 0.115·51-s + 0.544·53-s − 0.529·57-s − 0.838·59-s − 1.66·61-s − 0.143·63-s + 0.269·67-s + 0.293·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 8 T + p^{3} T^{2} \) |
| 11 | \( 1 - 20 T + p^{3} T^{2} \) |
| 13 | \( 1 + 22 T + p^{3} T^{2} \) |
| 17 | \( 1 - 14 T + p^{3} T^{2} \) |
| 19 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 23 | \( 1 + 56 T + p^{3} T^{2} \) |
| 29 | \( 1 + 154 T + p^{3} T^{2} \) |
| 31 | \( 1 - 160 T + p^{3} T^{2} \) |
| 37 | \( 1 - 162 T + p^{3} T^{2} \) |
| 41 | \( 1 + 390 T + p^{3} T^{2} \) |
| 43 | \( 1 + 388 T + p^{3} T^{2} \) |
| 47 | \( 1 - 544 T + p^{3} T^{2} \) |
| 53 | \( 1 - 210 T + p^{3} T^{2} \) |
| 59 | \( 1 + 380 T + p^{3} T^{2} \) |
| 61 | \( 1 + 794 T + p^{3} T^{2} \) |
| 67 | \( 1 - 148 T + p^{3} T^{2} \) |
| 71 | \( 1 + 840 T + p^{3} T^{2} \) |
| 73 | \( 1 + 858 T + p^{3} T^{2} \) |
| 79 | \( 1 - 144 T + p^{3} T^{2} \) |
| 83 | \( 1 + 316 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1098 T + p^{3} T^{2} \) |
| 97 | \( 1 + 994 T + p^{3} T^{2} \) |
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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
−9.858195039064007808083291812846, −9.152756785828852556381630175871, −7.937480225642516303265012801209, −7.04398509773126147377112777179, −6.17434168605816562953460698531, −5.26760482819788343188426144623, −4.18261496955300194405272257579, −3.02016264350841949893002920227, −1.45650826873907689563251131852, 0,
1.45650826873907689563251131852, 3.02016264350841949893002920227, 4.18261496955300194405272257579, 5.26760482819788343188426144623, 6.17434168605816562953460698531, 7.04398509773126147377112777179, 7.937480225642516303265012801209, 9.152756785828852556381630175871, 9.858195039064007808083291812846
