L(s) = 1 | − 1.59·2-s + 3-s + 0.548·4-s − 4.15·5-s − 1.59·6-s − 4.26·7-s + 2.31·8-s + 9-s + 6.63·10-s − 3.96·11-s + 0.548·12-s + 5.96·13-s + 6.80·14-s − 4.15·15-s − 4.79·16-s + 17-s − 1.59·18-s − 3.39·19-s − 2.27·20-s − 4.26·21-s + 6.32·22-s − 1.77·23-s + 2.31·24-s + 12.2·25-s − 9.52·26-s + 27-s − 2.33·28-s + ⋯ |
L(s) = 1 | − 1.12·2-s + 0.577·3-s + 0.274·4-s − 1.85·5-s − 0.651·6-s − 1.61·7-s + 0.819·8-s + 0.333·9-s + 2.09·10-s − 1.19·11-s + 0.158·12-s + 1.65·13-s + 1.81·14-s − 1.07·15-s − 1.19·16-s + 0.242·17-s − 0.376·18-s − 0.778·19-s − 0.509·20-s − 0.930·21-s + 1.34·22-s − 0.369·23-s + 0.473·24-s + 2.45·25-s − 1.86·26-s + 0.192·27-s − 0.441·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 79 | \( 1 + T \) |
good | 2 | \( 1 + 1.59T + 2T^{2} \) |
| 5 | \( 1 + 4.15T + 5T^{2} \) |
| 7 | \( 1 + 4.26T + 7T^{2} \) |
| 11 | \( 1 + 3.96T + 11T^{2} \) |
| 13 | \( 1 - 5.96T + 13T^{2} \) |
| 19 | \( 1 + 3.39T + 19T^{2} \) |
| 23 | \( 1 + 1.77T + 23T^{2} \) |
| 29 | \( 1 + 1.91T + 29T^{2} \) |
| 31 | \( 1 - 7.60T + 31T^{2} \) |
| 37 | \( 1 - 4.78T + 37T^{2} \) |
| 41 | \( 1 - 0.393T + 41T^{2} \) |
| 43 | \( 1 + 9.34T + 43T^{2} \) |
| 47 | \( 1 - 3.81T + 47T^{2} \) |
| 53 | \( 1 + 6.91T + 53T^{2} \) |
| 59 | \( 1 - 7.28T + 59T^{2} \) |
| 61 | \( 1 + 3.33T + 61T^{2} \) |
| 67 | \( 1 - 11.7T + 67T^{2} \) |
| 71 | \( 1 - 14.0T + 71T^{2} \) |
| 73 | \( 1 - 3.83T + 73T^{2} \) |
| 83 | \( 1 + 2.25T + 83T^{2} \) |
| 89 | \( 1 - 13.0T + 89T^{2} \) |
| 97 | \( 1 + 2.41T + 97T^{2} \) |
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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
−8.310690158649672457735814262627, −7.70663551498714019647622904819, −6.88425389965055606568915534401, −6.25745807673790245004452643618, −4.85698170253777769416989960209, −3.89271052492323125398536087817, −3.50907276406983638197952541271, −2.54770238688056858330348210569, −0.889809882043389140422425145954, 0,
0.889809882043389140422425145954, 2.54770238688056858330348210569, 3.50907276406983638197952541271, 3.89271052492323125398536087817, 4.85698170253777769416989960209, 6.25745807673790245004452643618, 6.88425389965055606568915534401, 7.70663551498714019647622904819, 8.310690158649672457735814262627