L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 5·7-s + 8-s + 9-s − 12-s − 2·13-s − 5·14-s + 16-s − 3·17-s + 18-s + 2·19-s + 5·21-s − 23-s − 24-s − 2·26-s − 27-s − 5·28-s + 3·29-s + 2·31-s + 32-s − 3·34-s + 36-s + 7·37-s + 2·38-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 1.88·7-s + 0.353·8-s + 1/3·9-s − 0.288·12-s − 0.554·13-s − 1.33·14-s + 1/4·16-s − 0.727·17-s + 0.235·18-s + 0.458·19-s + 1.09·21-s − 0.208·23-s − 0.204·24-s − 0.392·26-s − 0.192·27-s − 0.944·28-s + 0.557·29-s + 0.359·31-s + 0.176·32-s − 0.514·34-s + 1/6·36-s + 1.15·37-s + 0.324·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.548929737\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.548929737\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 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
−8.690946808893928521187437984821, −7.45524025254421600856967794267, −6.92553515487321769668775907099, −6.22289040003527569430337196463, −5.75057477662659755547102591322, −4.74332552864596727649359240601, −3.97447672206382845688374442524, −3.09969823459725034754533720693, −2.33576206955171057365724298047, −0.65187739673594516013772594551,
0.65187739673594516013772594551, 2.33576206955171057365724298047, 3.09969823459725034754533720693, 3.97447672206382845688374442524, 4.74332552864596727649359240601, 5.75057477662659755547102591322, 6.22289040003527569430337196463, 6.92553515487321769668775907099, 7.45524025254421600856967794267, 8.690946808893928521187437984821