L(s) = 1 | − 2·2-s + 3-s + 2·4-s + 5-s − 2·6-s + 9-s − 2·10-s − 6·11-s + 2·12-s − 3·13-s + 15-s − 4·16-s − 4·17-s − 2·18-s + 19-s + 2·20-s + 12·22-s − 4·23-s + 25-s + 6·26-s + 27-s − 8·29-s − 2·30-s + 31-s + 8·32-s − 6·33-s + 8·34-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 0.577·3-s + 4-s + 0.447·5-s − 0.816·6-s + 1/3·9-s − 0.632·10-s − 1.80·11-s + 0.577·12-s − 0.832·13-s + 0.258·15-s − 16-s − 0.970·17-s − 0.471·18-s + 0.229·19-s + 0.447·20-s + 2.55·22-s − 0.834·23-s + 1/5·25-s + 1.17·26-s + 0.192·27-s − 1.48·29-s − 0.365·30-s + 0.179·31-s + 1.41·32-s − 1.04·33-s + 1.37·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + p T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−9.748395957074705728171348053572, −9.245180260331893745789230775379, −8.190905413721799078782891747604, −7.73500063744226361021680192017, −6.88801796507275225595285396641, −5.54104550542340361135724955036, −4.45568069611345949977480819854, −2.71878989525937820035970356719, −1.94009500889479360979174607930, 0,
1.94009500889479360979174607930, 2.71878989525937820035970356719, 4.45568069611345949977480819854, 5.54104550542340361135724955036, 6.88801796507275225595285396641, 7.73500063744226361021680192017, 8.190905413721799078782891747604, 9.245180260331893745789230775379, 9.748395957074705728171348053572