| L(s) = 1 | − 2-s − 1.90·3-s + 4-s + 0.655·5-s + 1.90·6-s + 2.04·7-s − 8-s + 0.631·9-s − 0.655·10-s + 1.51·11-s − 1.90·12-s − 1.53·13-s − 2.04·14-s − 1.24·15-s + 16-s + 0.126·17-s − 0.631·18-s − 4.09·19-s + 0.655·20-s − 3.90·21-s − 1.51·22-s + 7.24·23-s + 1.90·24-s − 4.56·25-s + 1.53·26-s + 4.51·27-s + 2.04·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.10·3-s + 0.5·4-s + 0.293·5-s + 0.778·6-s + 0.774·7-s − 0.353·8-s + 0.210·9-s − 0.207·10-s + 0.455·11-s − 0.550·12-s − 0.426·13-s − 0.547·14-s − 0.322·15-s + 0.250·16-s + 0.0307·17-s − 0.148·18-s − 0.940·19-s + 0.146·20-s − 0.852·21-s − 0.322·22-s + 1.51·23-s + 0.389·24-s − 0.913·25-s + 0.301·26-s + 0.868·27-s + 0.387·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7891243366\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7891243366\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 269 | \( 1 - T \) |
| good | 3 | \( 1 + 1.90T + 3T^{2} \) |
| 5 | \( 1 - 0.655T + 5T^{2} \) |
| 7 | \( 1 - 2.04T + 7T^{2} \) |
| 11 | \( 1 - 1.51T + 11T^{2} \) |
| 13 | \( 1 + 1.53T + 13T^{2} \) |
| 17 | \( 1 - 0.126T + 17T^{2} \) |
| 19 | \( 1 + 4.09T + 19T^{2} \) |
| 23 | \( 1 - 7.24T + 23T^{2} \) |
| 29 | \( 1 - 8.20T + 29T^{2} \) |
| 31 | \( 1 - 1.17T + 31T^{2} \) |
| 37 | \( 1 - 7.34T + 37T^{2} \) |
| 41 | \( 1 + 4.61T + 41T^{2} \) |
| 43 | \( 1 - 8.19T + 43T^{2} \) |
| 47 | \( 1 - 7.49T + 47T^{2} \) |
| 53 | \( 1 - 11.1T + 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 + 2.46T + 61T^{2} \) |
| 67 | \( 1 - 6.04T + 67T^{2} \) |
| 71 | \( 1 - 3.67T + 71T^{2} \) |
| 73 | \( 1 + 12.2T + 73T^{2} \) |
| 79 | \( 1 + 4.60T + 79T^{2} \) |
| 83 | \( 1 + 0.483T + 83T^{2} \) |
| 89 | \( 1 - 4.00T + 89T^{2} \) |
| 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
−10.81331985073455630587419777216, −10.14087037249222735043670250416, −9.048599452245478761455112207660, −8.254601961328409481467015797144, −7.12031803104349283703542709383, −6.29566517806229637454684326185, −5.38603252494588016042016578685, −4.38730406642955465156931706862, −2.50292005003226576854470258159, −0.963312960833360907707787015655,
0.963312960833360907707787015655, 2.50292005003226576854470258159, 4.38730406642955465156931706862, 5.38603252494588016042016578685, 6.29566517806229637454684326185, 7.12031803104349283703542709383, 8.254601961328409481467015797144, 9.048599452245478761455112207660, 10.14087037249222735043670250416, 10.81331985073455630587419777216
