| L(s) = 1 | − 3·3-s + 6·9-s + 11-s + 2·13-s + 3·17-s + 5·19-s − 3·23-s − 9·27-s − 6·29-s − 31-s − 3·33-s + 5·37-s − 6·39-s + 10·41-s − 4·43-s − 47-s − 9·51-s + 9·53-s − 15·57-s + 3·59-s − 3·61-s + 11·67-s + 9·69-s − 16·71-s + 7·73-s + 11·79-s + 9·81-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 2·9-s + 0.301·11-s + 0.554·13-s + 0.727·17-s + 1.14·19-s − 0.625·23-s − 1.73·27-s − 1.11·29-s − 0.179·31-s − 0.522·33-s + 0.821·37-s − 0.960·39-s + 1.56·41-s − 0.609·43-s − 0.145·47-s − 1.26·51-s + 1.23·53-s − 1.98·57-s + 0.390·59-s − 0.384·61-s + 1.34·67-s + 1.08·69-s − 1.89·71-s + 0.819·73-s + 1.23·79-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.314273617\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.314273617\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + p T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 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
−15.92026975410728, −15.32990815831051, −14.60588340883713, −14.07525438416158, −13.24366559772656, −12.94175553862397, −12.13016307769938, −11.84341821393693, −11.33280519384736, −10.83397048389655, −10.26452800972355, −9.645939309163899, −9.214685609500232, −8.218208760570322, −7.511670071476873, −7.120449672738354, −6.228751575710689, −5.909241099256574, −5.369834917896218, −4.741458416627420, −3.962854074338414, −3.395914987402821, −2.157880851694797, −1.202177087494433, −0.6184840073684046,
0.6184840073684046, 1.202177087494433, 2.157880851694797, 3.395914987402821, 3.962854074338414, 4.741458416627420, 5.369834917896218, 5.909241099256574, 6.228751575710689, 7.120449672738354, 7.511670071476873, 8.218208760570322, 9.214685609500232, 9.645939309163899, 10.26452800972355, 10.83397048389655, 11.33280519384736, 11.84341821393693, 12.13016307769938, 12.94175553862397, 13.24366559772656, 14.07525438416158, 14.60588340883713, 15.32990815831051, 15.92026975410728
