| L(s) = 1 | + 1.61·2-s + 3-s + 0.618·4-s − 0.381·5-s + 1.61·6-s + 3·7-s − 2.23·8-s + 9-s − 0.618·10-s + 0.618·12-s + 6.23·13-s + 4.85·14-s − 0.381·15-s − 4.85·16-s − 0.618·17-s + 1.61·18-s − 0.854·19-s − 0.236·20-s + 3·21-s − 5.47·23-s − 2.23·24-s − 4.85·25-s + 10.0·26-s + 27-s + 1.85·28-s − 4.47·29-s − 0.618·30-s + ⋯ |
| L(s) = 1 | + 1.14·2-s + 0.577·3-s + 0.309·4-s − 0.170·5-s + 0.660·6-s + 1.13·7-s − 0.790·8-s + 0.333·9-s − 0.195·10-s + 0.178·12-s + 1.72·13-s + 1.29·14-s − 0.0986·15-s − 1.21·16-s − 0.149·17-s + 0.381·18-s − 0.195·19-s − 0.0527·20-s + 0.654·21-s − 1.14·23-s − 0.456·24-s − 0.970·25-s + 1.97·26-s + 0.192·27-s + 0.350·28-s − 0.830·29-s − 0.112·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.721153464\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.721153464\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 1.61T + 2T^{2} \) |
| 5 | \( 1 + 0.381T + 5T^{2} \) |
| 7 | \( 1 - 3T + 7T^{2} \) |
| 13 | \( 1 - 6.23T + 13T^{2} \) |
| 17 | \( 1 + 0.618T + 17T^{2} \) |
| 19 | \( 1 + 0.854T + 19T^{2} \) |
| 23 | \( 1 + 5.47T + 23T^{2} \) |
| 29 | \( 1 + 4.47T + 29T^{2} \) |
| 31 | \( 1 + 3.85T + 31T^{2} \) |
| 37 | \( 1 + 4.23T + 37T^{2} \) |
| 41 | \( 1 - 5.94T + 41T^{2} \) |
| 43 | \( 1 - 1.76T + 43T^{2} \) |
| 47 | \( 1 + 0.618T + 47T^{2} \) |
| 53 | \( 1 + 7.38T + 53T^{2} \) |
| 59 | \( 1 + 5.32T + 59T^{2} \) |
| 61 | \( 1 - 1.14T + 61T^{2} \) |
| 67 | \( 1 - 10.5T + 67T^{2} \) |
| 71 | \( 1 - 14.5T + 71T^{2} \) |
| 73 | \( 1 - 1.23T + 73T^{2} \) |
| 79 | \( 1 - 0.527T + 79T^{2} \) |
| 83 | \( 1 + 12.7T + 83T^{2} \) |
| 89 | \( 1 - 9.47T + 89T^{2} \) |
| 97 | \( 1 - 15.0T + 97T^{2} \) |
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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
−11.50398749116343667739074270250, −10.89581250907366373252466324033, −9.439131134488799967977287715585, −8.513070950631042094322441331278, −7.79526601890347810468062167552, −6.33312158733603880861521065816, −5.41772466495137110744896176397, −4.21280858197488992738118398970, −3.58177656039584076646750252854, −1.92517922199044961747531354354,
1.92517922199044961747531354354, 3.58177656039584076646750252854, 4.21280858197488992738118398970, 5.41772466495137110744896176397, 6.33312158733603880861521065816, 7.79526601890347810468062167552, 8.513070950631042094322441331278, 9.439131134488799967977287715585, 10.89581250907366373252466324033, 11.50398749116343667739074270250
