| L(s) = 1 | − 2.52·2-s + 3-s + 4.37·4-s − 0.338·5-s − 2.52·6-s + 7-s − 5.98·8-s + 9-s + 0.853·10-s + 5.96·11-s + 4.37·12-s − 1.38·13-s − 2.52·14-s − 0.338·15-s + 6.36·16-s + 1.72·17-s − 2.52·18-s − 0.951·19-s − 1.47·20-s + 21-s − 15.0·22-s + 1.87·23-s − 5.98·24-s − 4.88·25-s + 3.49·26-s + 27-s + 4.37·28-s + ⋯ |
| L(s) = 1 | − 1.78·2-s + 0.577·3-s + 2.18·4-s − 0.151·5-s − 1.03·6-s + 0.377·7-s − 2.11·8-s + 0.333·9-s + 0.269·10-s + 1.79·11-s + 1.26·12-s − 0.384·13-s − 0.674·14-s − 0.0873·15-s + 1.59·16-s + 0.417·17-s − 0.594·18-s − 0.218·19-s − 0.330·20-s + 0.218·21-s − 3.21·22-s + 0.391·23-s − 1.22·24-s − 0.977·25-s + 0.685·26-s + 0.192·27-s + 0.826·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.197598715\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.197598715\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 191 | \( 1 - T \) |
| good | 2 | \( 1 + 2.52T + 2T^{2} \) |
| 5 | \( 1 + 0.338T + 5T^{2} \) |
| 11 | \( 1 - 5.96T + 11T^{2} \) |
| 13 | \( 1 + 1.38T + 13T^{2} \) |
| 17 | \( 1 - 1.72T + 17T^{2} \) |
| 19 | \( 1 + 0.951T + 19T^{2} \) |
| 23 | \( 1 - 1.87T + 23T^{2} \) |
| 29 | \( 1 - 1.24T + 29T^{2} \) |
| 31 | \( 1 + 3.41T + 31T^{2} \) |
| 37 | \( 1 - 4.79T + 37T^{2} \) |
| 41 | \( 1 - 10.9T + 41T^{2} \) |
| 43 | \( 1 - 4.73T + 43T^{2} \) |
| 47 | \( 1 + 0.146T + 47T^{2} \) |
| 53 | \( 1 + 5.65T + 53T^{2} \) |
| 59 | \( 1 - 3.20T + 59T^{2} \) |
| 61 | \( 1 - 4.89T + 61T^{2} \) |
| 67 | \( 1 + 14.4T + 67T^{2} \) |
| 71 | \( 1 - 13.4T + 71T^{2} \) |
| 73 | \( 1 + 8.54T + 73T^{2} \) |
| 79 | \( 1 - 6.16T + 79T^{2} \) |
| 83 | \( 1 - 1.60T + 83T^{2} \) |
| 89 | \( 1 + 7.67T + 89T^{2} \) |
| 97 | \( 1 - 16.6T + 97T^{2} \) |
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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
−8.595315793427288208864440637119, −7.76079016081020933493763695908, −7.41689967705384809217744170692, −6.57124068339826730095308018772, −5.90103839448640412418204358058, −4.47725783027265195644236610866, −3.64830331871282509143888859527, −2.53178190598542350126397193546, −1.67475922868837251599884912487, −0.846437027375014445307395726201,
0.846437027375014445307395726201, 1.67475922868837251599884912487, 2.53178190598542350126397193546, 3.64830331871282509143888859527, 4.47725783027265195644236610866, 5.90103839448640412418204358058, 6.57124068339826730095308018772, 7.41689967705384809217744170692, 7.76079016081020933493763695908, 8.595315793427288208864440637119
