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
\(L(1)\) |
\(\approx\) |
\(0.4133357624\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4133357624\) |
\(L(\frac{3}{2})\) |
|
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
−10.28951334699592918656639939613, −9.683061530228168119703155484024, −8.557316152456589603921207372025, −7.84525423583880781096311301042, −7.36558585620663223964460201996, −6.05889584766761249242853724761, −5.15374272249316546367796432469, −3.83572932104411065037140648432, −2.24492990728449161387935433816, −0.66746462687447569978881815058,
0.66746462687447569978881815058, 2.24492990728449161387935433816, 3.83572932104411065037140648432, 5.15374272249316546367796432469, 6.05889584766761249242853724761, 7.36558585620663223964460201996, 7.84525423583880781096311301042, 8.557316152456589603921207372025, 9.683061530228168119703155484024, 10.28951334699592918656639939613