L(s) = 1 | + 3-s − 2·9-s − 3·11-s + 2·13-s + 3·17-s + 19-s − 3·23-s − 5·27-s − 6·29-s + 7·31-s − 3·33-s + 37-s + 2·39-s − 6·41-s + 4·43-s − 9·47-s + 3·51-s − 3·53-s + 57-s − 9·59-s + 61-s + 7·67-s − 3·69-s − 73-s − 13·79-s + 81-s + 12·83-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 2/3·9-s − 0.904·11-s + 0.554·13-s + 0.727·17-s + 0.229·19-s − 0.625·23-s − 0.962·27-s − 1.11·29-s + 1.25·31-s − 0.522·33-s + 0.164·37-s + 0.320·39-s − 0.937·41-s + 0.609·43-s − 1.31·47-s + 0.420·51-s − 0.412·53-s + 0.132·57-s − 1.17·59-s + 0.128·61-s + 0.855·67-s − 0.361·69-s − 0.117·73-s − 1.46·79-s + 1/9·81-s + 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−8.131003372691400825176391323269, −7.38711655989492333958567538252, −6.38882457631906124699452637079, −5.69270808305625472851964688008, −5.05757201954760887014741309186, −3.99318106967751438511095059887, −3.20659781016690415261077023819, −2.56336116719605450004345950124, −1.48712082202268856378613323215, 0,
1.48712082202268856378613323215, 2.56336116719605450004345950124, 3.20659781016690415261077023819, 3.99318106967751438511095059887, 5.05757201954760887014741309186, 5.69270808305625472851964688008, 6.38882457631906124699452637079, 7.38711655989492333958567538252, 8.131003372691400825176391323269