| L(s) = 1 | + 9·9-s + 10·13-s + 30·17-s − 42·29-s − 70·37-s + 18·41-s + 49·49-s + 90·53-s + 22·61-s + 110·73-s + 81·81-s − 78·89-s − 130·97-s + 198·101-s + 182·109-s + 30·113-s + 90·117-s + ⋯ |
| L(s) = 1 | + 9-s + 0.769·13-s + 1.76·17-s − 1.44·29-s − 1.89·37-s + 0.439·41-s + 49-s + 1.69·53-s + 0.360·61-s + 1.50·73-s + 81-s − 0.876·89-s − 1.34·97-s + 1.96·101-s + 1.66·109-s + 0.265·113-s + 0.769·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.409960127\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.409960127\) |
| \(L(2)\) |
|
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 | \( ( 1 - p T )( 1 + p T ) \) |
| 7 | \( ( 1 - p T )( 1 + p T ) \) |
| 11 | \( ( 1 - p T )( 1 + p T ) \) |
| 13 | \( 1 - 10 T + p^{2} T^{2} \) |
| 17 | \( 1 - 30 T + p^{2} T^{2} \) |
| 19 | \( ( 1 - p T )( 1 + p T ) \) |
| 23 | \( ( 1 - p T )( 1 + p T ) \) |
| 29 | \( 1 + 42 T + p^{2} T^{2} \) |
| 31 | \( ( 1 - p T )( 1 + p T ) \) |
| 37 | \( 1 + 70 T + p^{2} T^{2} \) |
| 41 | \( 1 - 18 T + p^{2} T^{2} \) |
| 43 | \( ( 1 - p T )( 1 + p T ) \) |
| 47 | \( ( 1 - p T )( 1 + p T ) \) |
| 53 | \( 1 - 90 T + p^{2} T^{2} \) |
| 59 | \( ( 1 - p T )( 1 + p T ) \) |
| 61 | \( 1 - 22 T + p^{2} T^{2} \) |
| 67 | \( ( 1 - p T )( 1 + p T ) \) |
| 71 | \( ( 1 - p T )( 1 + p T ) \) |
| 73 | \( 1 - 110 T + p^{2} T^{2} \) |
| 79 | \( ( 1 - p T )( 1 + p T ) \) |
| 83 | \( ( 1 - p T )( 1 + p T ) \) |
| 89 | \( 1 + 78 T + p^{2} T^{2} \) |
| 97 | \( 1 + 130 T + p^{2} T^{2} \) |
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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
−9.266111874409491464702572109444, −8.421604999422886110856437951095, −7.52811605302068276303622600329, −6.98267977965120420060844208806, −5.85879046004519036359983248707, −5.21972899574520890684064063109, −3.99487247109607255108489912048, −3.40584005583173808145100182005, −1.93575569798055937544859014171, −0.924997376009097049877988639573,
0.924997376009097049877988639573, 1.93575569798055937544859014171, 3.40584005583173808145100182005, 3.99487247109607255108489912048, 5.21972899574520890684064063109, 5.85879046004519036359983248707, 6.98267977965120420060844208806, 7.52811605302068276303622600329, 8.421604999422886110856437951095, 9.266111874409491464702572109444
