L(s) = 1 | − 3-s − 4.30·5-s − 7-s + 9-s − 1.60·11-s − 1.30·13-s + 4.30·15-s − 7.60·17-s − 7.60·19-s + 21-s − 23-s + 13.5·25-s − 27-s + 1.60·29-s + 3.60·31-s + 1.60·33-s + 4.30·35-s − 1.60·37-s + 1.30·39-s − 3.60·41-s + 0.697·43-s − 4.30·45-s + 2.60·47-s + 49-s + 7.60·51-s − 4.90·53-s + 6.90·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.92·5-s − 0.377·7-s + 0.333·9-s − 0.484·11-s − 0.361·13-s + 1.11·15-s − 1.84·17-s − 1.74·19-s + 0.218·21-s − 0.208·23-s + 2.70·25-s − 0.192·27-s + 0.298·29-s + 0.647·31-s + 0.279·33-s + 0.727·35-s − 0.263·37-s + 0.208·39-s − 0.563·41-s + 0.106·43-s − 0.641·45-s + 0.380·47-s + 0.142·49-s + 1.06·51-s − 0.674·53-s + 0.931·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2940342573\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2940342573\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 + 4.30T + 5T^{2} \) |
| 11 | \( 1 + 1.60T + 11T^{2} \) |
| 13 | \( 1 + 1.30T + 13T^{2} \) |
| 17 | \( 1 + 7.60T + 17T^{2} \) |
| 19 | \( 1 + 7.60T + 19T^{2} \) |
| 29 | \( 1 - 1.60T + 29T^{2} \) |
| 31 | \( 1 - 3.60T + 31T^{2} \) |
| 37 | \( 1 + 1.60T + 37T^{2} \) |
| 41 | \( 1 + 3.60T + 41T^{2} \) |
| 43 | \( 1 - 0.697T + 43T^{2} \) |
| 47 | \( 1 - 2.60T + 47T^{2} \) |
| 53 | \( 1 + 4.90T + 53T^{2} \) |
| 59 | \( 1 + 6.90T + 59T^{2} \) |
| 61 | \( 1 + 5.90T + 61T^{2} \) |
| 67 | \( 1 - 9.69T + 67T^{2} \) |
| 71 | \( 1 - 2.09T + 71T^{2} \) |
| 73 | \( 1 - 11T + 73T^{2} \) |
| 79 | \( 1 + 15.4T + 79T^{2} \) |
| 83 | \( 1 - 1.78T + 83T^{2} \) |
| 89 | \( 1 - 2.51T + 89T^{2} \) |
| 97 | \( 1 - 18.2T + 97T^{2} \) |
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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
−8.933137885356858032540212937031, −8.401575175771316064170692889889, −7.60766984145757659779817334744, −6.81708587306179222425004814557, −6.26691375538174925191032995529, −4.75506945257044346815865649897, −4.44572679667145837816937068077, −3.50548322087512858493652973024, −2.32318375918765043319228948734, −0.35108444683536224189146360872,
0.35108444683536224189146360872, 2.32318375918765043319228948734, 3.50548322087512858493652973024, 4.44572679667145837816937068077, 4.75506945257044346815865649897, 6.26691375538174925191032995529, 6.81708587306179222425004814557, 7.60766984145757659779817334744, 8.401575175771316064170692889889, 8.933137885356858032540212937031