| L(s) = 1 | − 2.06·2-s − 3-s + 2.26·4-s − 0.608·5-s + 2.06·6-s − 0.546·8-s + 9-s + 1.25·10-s − 1.10·11-s − 2.26·12-s + 1.84·13-s + 0.608·15-s − 3.40·16-s + 5.88·17-s − 2.06·18-s + 1.08·19-s − 1.37·20-s + 2.27·22-s + 23-s + 0.546·24-s − 4.62·25-s − 3.81·26-s − 27-s − 0.804·29-s − 1.25·30-s + 8.77·31-s + 8.11·32-s + ⋯ |
| L(s) = 1 | − 1.46·2-s − 0.577·3-s + 1.13·4-s − 0.272·5-s + 0.843·6-s − 0.193·8-s + 0.333·9-s + 0.397·10-s − 0.332·11-s − 0.653·12-s + 0.512·13-s + 0.157·15-s − 0.850·16-s + 1.42·17-s − 0.486·18-s + 0.249·19-s − 0.308·20-s + 0.485·22-s + 0.208·23-s + 0.111·24-s − 0.925·25-s − 0.748·26-s − 0.192·27-s − 0.149·29-s − 0.229·30-s + 1.57·31-s + 1.43·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6376287831\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6376287831\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 + 2.06T + 2T^{2} \) |
| 5 | \( 1 + 0.608T + 5T^{2} \) |
| 11 | \( 1 + 1.10T + 11T^{2} \) |
| 13 | \( 1 - 1.84T + 13T^{2} \) |
| 17 | \( 1 - 5.88T + 17T^{2} \) |
| 19 | \( 1 - 1.08T + 19T^{2} \) |
| 29 | \( 1 + 0.804T + 29T^{2} \) |
| 31 | \( 1 - 8.77T + 31T^{2} \) |
| 37 | \( 1 - 9.11T + 37T^{2} \) |
| 41 | \( 1 + 9.79T + 41T^{2} \) |
| 43 | \( 1 - 5.23T + 43T^{2} \) |
| 47 | \( 1 + 5.90T + 47T^{2} \) |
| 53 | \( 1 - 4.51T + 53T^{2} \) |
| 59 | \( 1 + 7.40T + 59T^{2} \) |
| 61 | \( 1 - 5.02T + 61T^{2} \) |
| 67 | \( 1 - 2.06T + 67T^{2} \) |
| 71 | \( 1 - 4.30T + 71T^{2} \) |
| 73 | \( 1 + 1.15T + 73T^{2} \) |
| 79 | \( 1 - 15.1T + 79T^{2} \) |
| 83 | \( 1 + 10.7T + 83T^{2} \) |
| 89 | \( 1 + 7.92T + 89T^{2} \) |
| 97 | \( 1 + 17.0T + 97T^{2} \) |
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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
−8.432498384073886882653970271272, −8.024019338753971344339799622212, −7.39681688782931761220270723138, −6.56180757724017849512218296588, −5.79071180708948986554678472776, −4.89095606813865919867062982248, −3.90755222533992550805640249743, −2.77984710864393777247253622971, −1.51077818719431514595773456197, −0.65666206084317941630737210210,
0.65666206084317941630737210210, 1.51077818719431514595773456197, 2.77984710864393777247253622971, 3.90755222533992550805640249743, 4.89095606813865919867062982248, 5.79071180708948986554678472776, 6.56180757724017849512218296588, 7.39681688782931761220270723138, 8.024019338753971344339799622212, 8.432498384073886882653970271272
