L(s) = 1 | − 2.01·2-s + 3-s + 2.05·4-s − 0.898·5-s − 2.01·6-s − 4.86·7-s − 0.101·8-s + 9-s + 1.80·10-s − 0.654·11-s + 2.05·12-s − 4.49·13-s + 9.78·14-s − 0.898·15-s − 3.89·16-s − 17-s − 2.01·18-s + 4.80·19-s − 1.84·20-s − 4.86·21-s + 1.31·22-s + 7.89·23-s − 0.101·24-s − 4.19·25-s + 9.03·26-s + 27-s − 9.96·28-s + ⋯ |
L(s) = 1 | − 1.42·2-s + 0.577·3-s + 1.02·4-s − 0.401·5-s − 0.821·6-s − 1.83·7-s − 0.0359·8-s + 0.333·9-s + 0.572·10-s − 0.197·11-s + 0.591·12-s − 1.24·13-s + 2.61·14-s − 0.232·15-s − 0.974·16-s − 0.242·17-s − 0.474·18-s + 1.10·19-s − 0.412·20-s − 1.06·21-s + 0.280·22-s + 1.64·23-s − 0.0207·24-s − 0.838·25-s + 1.77·26-s + 0.192·27-s − 1.88·28-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 | \( 1 + 2.01T + 2T^{2} \) |
| 5 | \( 1 + 0.898T + 5T^{2} \) |
| 7 | \( 1 + 4.86T + 7T^{2} \) |
| 11 | \( 1 + 0.654T + 11T^{2} \) |
| 13 | \( 1 + 4.49T + 13T^{2} \) |
| 19 | \( 1 - 4.80T + 19T^{2} \) |
| 23 | \( 1 - 7.89T + 23T^{2} \) |
| 29 | \( 1 - 2.96T + 29T^{2} \) |
| 31 | \( 1 - 6.47T + 31T^{2} \) |
| 37 | \( 1 - 9.04T + 37T^{2} \) |
| 41 | \( 1 + 11.5T + 41T^{2} \) |
| 43 | \( 1 - 10.4T + 43T^{2} \) |
| 47 | \( 1 - 3.33T + 47T^{2} \) |
| 53 | \( 1 + 4.23T + 53T^{2} \) |
| 59 | \( 1 - 5.44T + 59T^{2} \) |
| 61 | \( 1 + 13.4T + 61T^{2} \) |
| 67 | \( 1 + 6.76T + 67T^{2} \) |
| 71 | \( 1 - 4.22T + 71T^{2} \) |
| 73 | \( 1 - 8.74T + 73T^{2} \) |
| 83 | \( 1 + 17.0T + 83T^{2} \) |
| 89 | \( 1 - 1.07T + 89T^{2} \) |
| 97 | \( 1 - 6.15T + 97T^{2} \) |
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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
−8.091947827332151418959892886131, −7.47676015569949652918759659223, −6.98243129689681568142699972106, −6.28158593188112967723612678005, −5.04850732038958243757560099634, −4.09245259112135979879805095032, −2.95388102314558527297743071023, −2.60527610113689175910138240936, −1.03754474500517492511232240094, 0,
1.03754474500517492511232240094, 2.60527610113689175910138240936, 2.95388102314558527297743071023, 4.09245259112135979879805095032, 5.04850732038958243757560099634, 6.28158593188112967723612678005, 6.98243129689681568142699972106, 7.47676015569949652918759659223, 8.091947827332151418959892886131