| L(s) = 1 | + 0.891·2-s + 1.47·3-s − 0.205·4-s + 1.31·6-s − 1.70·7-s − 1.07·8-s + 1.18·9-s − 0.303·12-s − 1.51·14-s − 0.752·16-s + 1.86·17-s + 1.05·18-s − 1.96·19-s − 2.51·21-s − 0.547·23-s − 1.58·24-s + 25-s + 0.272·27-s + 0.349·28-s − 1.20·29-s + 1.86·31-s + 0.403·32-s + 1.66·34-s − 0.243·36-s − 1.75·38-s − 2.24·42-s + 0.184·43-s + ⋯ |
| L(s) = 1 | + 0.891·2-s + 1.47·3-s − 0.205·4-s + 1.31·6-s − 1.70·7-s − 1.07·8-s + 1.18·9-s − 0.303·12-s − 1.51·14-s − 0.752·16-s + 1.86·17-s + 1.05·18-s − 1.96·19-s − 2.51·21-s − 0.547·23-s − 1.58·24-s + 25-s + 0.272·27-s + 0.349·28-s − 1.20·29-s + 1.86·31-s + 0.403·32-s + 1.66·34-s − 0.243·36-s − 1.75·38-s − 2.24·42-s + 0.184·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 383 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 383 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.431067236\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.431067236\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 383 | \( 1 - T \) |
| good | 2 | \( 1 - 0.891T + T^{2} \) |
| 3 | \( 1 - 1.47T + T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 + 1.70T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - 1.86T + T^{2} \) |
| 19 | \( 1 + 1.96T + T^{2} \) |
| 23 | \( 1 + 0.547T + T^{2} \) |
| 29 | \( 1 + 1.20T + T^{2} \) |
| 31 | \( 1 - 1.86T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - 0.184T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + 0.547T + T^{2} \) |
| 71 | \( 1 - 0.184T + T^{2} \) |
| 73 | \( 1 + 0.547T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−12.12478836322166737712407082904, −10.30990921324437962391350196378, −9.623861615623269093766998701375, −8.866471556374795387314424262367, −8.011441433238300186406678981185, −6.68519497218131499508559325096, −5.79682017306189312189117808696, −4.22476005282874989607483117241, −3.40525931811136009959446897213, −2.68118171349689725668581355238,
2.68118171349689725668581355238, 3.40525931811136009959446897213, 4.22476005282874989607483117241, 5.79682017306189312189117808696, 6.68519497218131499508559325096, 8.011441433238300186406678981185, 8.866471556374795387314424262367, 9.623861615623269093766998701375, 10.30990921324437962391350196378, 12.12478836322166737712407082904
