L(s) = 1 | + 3-s − 5-s + 9-s − 13-s − 15-s + 3·17-s − 4·23-s − 4·25-s + 27-s − 29-s − 37-s − 39-s − 9·41-s − 4·43-s − 45-s + 4·47-s − 7·49-s + 3·51-s + 3·53-s − 12·59-s + 14·61-s + 65-s − 16·67-s − 4·69-s + 12·71-s + 10·73-s − 4·75-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.277·13-s − 0.258·15-s + 0.727·17-s − 0.834·23-s − 4/5·25-s + 0.192·27-s − 0.185·29-s − 0.164·37-s − 0.160·39-s − 1.40·41-s − 0.609·43-s − 0.149·45-s + 0.583·47-s − 49-s + 0.420·51-s + 0.412·53-s − 1.56·59-s + 1.79·61-s + 0.124·65-s − 1.95·67-s − 0.481·69-s + 1.42·71-s + 1.17·73-s − 0.461·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 5 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−7.889098103752808381954308797822, −7.19067861341124758894675753831, −6.42564540364361003026495586154, −5.56091761933383491736031069656, −4.78710296048331983730033121325, −3.88572355638926093846542583543, −3.36136672411098217913342621337, −2.36546597999485353505524324660, −1.45153886197015474009231852296, 0,
1.45153886197015474009231852296, 2.36546597999485353505524324660, 3.36136672411098217913342621337, 3.88572355638926093846542583543, 4.78710296048331983730033121325, 5.56091761933383491736031069656, 6.42564540364361003026495586154, 7.19067861341124758894675753831, 7.889098103752808381954308797822