| L(s) = 1 | + 1.76·2-s − 3-s + 1.11·4-s − 2.54·5-s − 1.76·6-s − 4.97·7-s − 1.56·8-s + 9-s − 4.49·10-s + 0.283·11-s − 1.11·12-s − 0.170·13-s − 8.78·14-s + 2.54·15-s − 4.98·16-s + 5.92·17-s + 1.76·18-s + 4.65·19-s − 2.84·20-s + 4.97·21-s + 0.499·22-s + 23-s + 1.56·24-s + 1.49·25-s − 0.301·26-s − 27-s − 5.54·28-s + ⋯ |
| L(s) = 1 | + 1.24·2-s − 0.577·3-s + 0.557·4-s − 1.13·5-s − 0.720·6-s − 1.88·7-s − 0.552·8-s + 0.333·9-s − 1.42·10-s + 0.0853·11-s − 0.321·12-s − 0.0474·13-s − 2.34·14-s + 0.657·15-s − 1.24·16-s + 1.43·17-s + 0.415·18-s + 1.06·19-s − 0.635·20-s + 1.08·21-s + 0.106·22-s + 0.208·23-s + 0.318·24-s + 0.298·25-s − 0.0591·26-s − 0.192·27-s − 1.04·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.294243804\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.294243804\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - 1.76T + 2T^{2} \) |
| 5 | \( 1 + 2.54T + 5T^{2} \) |
| 7 | \( 1 + 4.97T + 7T^{2} \) |
| 11 | \( 1 - 0.283T + 11T^{2} \) |
| 13 | \( 1 + 0.170T + 13T^{2} \) |
| 17 | \( 1 - 5.92T + 17T^{2} \) |
| 19 | \( 1 - 4.65T + 19T^{2} \) |
| 31 | \( 1 - 8.96T + 31T^{2} \) |
| 37 | \( 1 - 0.201T + 37T^{2} \) |
| 41 | \( 1 + 6.77T + 41T^{2} \) |
| 43 | \( 1 + 2.97T + 43T^{2} \) |
| 47 | \( 1 - 7.64T + 47T^{2} \) |
| 53 | \( 1 - 2.22T + 53T^{2} \) |
| 59 | \( 1 + 15.2T + 59T^{2} \) |
| 61 | \( 1 - 1.08T + 61T^{2} \) |
| 67 | \( 1 + 4.78T + 67T^{2} \) |
| 71 | \( 1 - 15.8T + 71T^{2} \) |
| 73 | \( 1 - 5.09T + 73T^{2} \) |
| 79 | \( 1 - 15.8T + 79T^{2} \) |
| 83 | \( 1 + 4.95T + 83T^{2} \) |
| 89 | \( 1 + 3.87T + 89T^{2} \) |
| 97 | \( 1 + 16.0T + 97T^{2} \) |
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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
−9.426124612088534037354615260845, −8.209469302944268024167528513970, −7.26547980047807790073758632673, −6.59191071308050078796221851753, −5.86987028840612654967664026185, −5.13785191375743560297004121043, −4.13087181808763572791256499757, −3.42263597594724536904888323179, −2.96220268822470293871484039842, −0.62563766928077496055130184787,
0.62563766928077496055130184787, 2.96220268822470293871484039842, 3.42263597594724536904888323179, 4.13087181808763572791256499757, 5.13785191375743560297004121043, 5.86987028840612654967664026185, 6.59191071308050078796221851753, 7.26547980047807790073758632673, 8.209469302944268024167528513970, 9.426124612088534037354615260845
