L(s) = 1 | − 5-s + 11-s + 2·17-s − 2·19-s + 8·23-s + 25-s + 4·29-s + 4·31-s + 2·37-s − 12·41-s + 8·43-s − 4·47-s − 7·49-s + 10·53-s − 55-s + 4·59-s + 6·61-s − 12·67-s + 16·73-s + 6·79-s − 2·83-s − 2·85-s + 18·89-s + 2·95-s + 2·97-s + 101-s + 103-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 0.301·11-s + 0.485·17-s − 0.458·19-s + 1.66·23-s + 1/5·25-s + 0.742·29-s + 0.718·31-s + 0.328·37-s − 1.87·41-s + 1.21·43-s − 0.583·47-s − 49-s + 1.37·53-s − 0.134·55-s + 0.520·59-s + 0.768·61-s − 1.46·67-s + 1.87·73-s + 0.675·79-s − 0.219·83-s − 0.216·85-s + 1.90·89-s + 0.205·95-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 334620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 334620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.105400134\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.105400134\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 7 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−12.47614385946017, −12.12141976660976, −11.72149226193307, −11.27200152243719, −10.74709917270220, −10.40767648451705, −9.862844347200100, −9.374108338734313, −8.909950470381526, −8.359759300068019, −8.148601908800536, −7.463642335526731, −6.975083079868059, −6.635441317278552, −6.134569948205288, −5.477686355084063, −4.923345423633974, −4.645646708420664, −3.956264011402052, −3.428754587012752, −3.003573988490372, −2.389900489917604, −1.706846911488259, −0.9531483597292837, −0.5617683400252486,
0.5617683400252486, 0.9531483597292837, 1.706846911488259, 2.389900489917604, 3.003573988490372, 3.428754587012752, 3.956264011402052, 4.645646708420664, 4.923345423633974, 5.477686355084063, 6.134569948205288, 6.635441317278552, 6.975083079868059, 7.463642335526731, 8.148601908800536, 8.359759300068019, 8.909950470381526, 9.374108338734313, 9.862844347200100, 10.40767648451705, 10.74709917270220, 11.27200152243719, 11.72149226193307, 12.12141976660976, 12.47614385946017