| L(s) = 1 | − 0.445·2-s − 0.801·4-s + 1.24·5-s + 0.801·8-s + 9-s − 0.554·10-s − 1.80·11-s + 0.445·16-s − 1.80·17-s − 0.445·18-s − 0.445·19-s − 20-s + 0.801·22-s + 0.554·25-s − 0.445·29-s + 1.24·31-s − 32-s + 0.801·34-s − 0.801·36-s − 1.80·37-s + 0.198·38-s + 40-s + 1.24·43-s + 1.44·44-s + 1.24·45-s − 0.445·47-s + 49-s + ⋯ |
| L(s) = 1 | − 0.445·2-s − 0.801·4-s + 1.24·5-s + 0.801·8-s + 9-s − 0.554·10-s − 1.80·11-s + 0.445·16-s − 1.80·17-s − 0.445·18-s − 0.445·19-s − 20-s + 0.801·22-s + 0.554·25-s − 0.445·29-s + 1.24·31-s − 32-s + 0.801·34-s − 0.801·36-s − 1.80·37-s + 0.198·38-s + 40-s + 1.24·43-s + 1.44·44-s + 1.24·45-s − 0.445·47-s + 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5222314318\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5222314318\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 151 | \( 1 - T \) |
| good | 2 | \( 1 + 0.445T + T^{2} \) |
| 3 | \( 1 - T^{2} \) |
| 5 | \( 1 - 1.24T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + 1.80T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + 1.80T + T^{2} \) |
| 19 | \( 1 + 0.445T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + 0.445T + T^{2} \) |
| 31 | \( 1 - 1.24T + T^{2} \) |
| 37 | \( 1 + 1.80T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - 1.24T + T^{2} \) |
| 47 | \( 1 + 0.445T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - 1.24T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + 0.445T + T^{2} \) |
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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
−13.34556557275344869968800790456, −12.68200101760350932318631160875, −10.68997024797752278504909682491, −10.19852870201252687896166884705, −9.251703683052829018409362241139, −8.247752176285942870731625179537, −6.95499951952608680352810802672, −5.48609729130887863451926607972, −4.45516567445391320591192585649, −2.17595385646999302389858797878,
2.17595385646999302389858797878, 4.45516567445391320591192585649, 5.48609729130887863451926607972, 6.95499951952608680352810802672, 8.247752176285942870731625179537, 9.251703683052829018409362241139, 10.19852870201252687896166884705, 10.68997024797752278504909682491, 12.68200101760350932318631160875, 13.34556557275344869968800790456
