L(s) = 1 | + 3-s + 5-s − 7-s + 9-s − 3.12·11-s − 5.12·13-s + 15-s + 2·17-s + 7.12·19-s − 21-s + 25-s + 27-s + 2·29-s + 3.12·31-s − 3.12·33-s − 35-s + 2·37-s − 5.12·39-s + 2·41-s − 6.24·43-s + 45-s + 49-s + 2·51-s + 11.3·53-s − 3.12·55-s + 7.12·57-s − 4·59-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.377·7-s + 0.333·9-s − 0.941·11-s − 1.42·13-s + 0.258·15-s + 0.485·17-s + 1.63·19-s − 0.218·21-s + 0.200·25-s + 0.192·27-s + 0.371·29-s + 0.560·31-s − 0.543·33-s − 0.169·35-s + 0.328·37-s − 0.820·39-s + 0.312·41-s − 0.952·43-s + 0.149·45-s + 0.142·49-s + 0.280·51-s + 1.56·53-s − 0.421·55-s + 0.943·57-s − 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.358603388\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.358603388\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 + 3.12T + 11T^{2} \) |
| 13 | \( 1 + 5.12T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 7.12T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 3.12T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 6.24T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 11.3T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 8.24T + 61T^{2} \) |
| 67 | \( 1 - 14.2T + 67T^{2} \) |
| 71 | \( 1 + 3.12T + 71T^{2} \) |
| 73 | \( 1 - 6.87T + 73T^{2} \) |
| 79 | \( 1 + 14.2T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 - 16.2T + 89T^{2} \) |
| 97 | \( 1 - 13.1T + 97T^{2} \) |
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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
−7.85257014440105896196143200969, −7.40686209308061927365299567961, −6.74795038615466565766836338760, −5.72612488907770516193443106055, −5.17414670875941747052104153232, −4.47849591494988199274333261964, −3.28642970886119721594219896259, −2.81473115612901459576922324511, −2.01792068801978651513931792596, −0.75264118017419226924907195799,
0.75264118017419226924907195799, 2.01792068801978651513931792596, 2.81473115612901459576922324511, 3.28642970886119721594219896259, 4.47849591494988199274333261964, 5.17414670875941747052104153232, 5.72612488907770516193443106055, 6.74795038615466565766836338760, 7.40686209308061927365299567961, 7.85257014440105896196143200969