L(s) = 1 | + 2.03·2-s − 3.04·3-s + 2.16·4-s + 5-s − 6.20·6-s + 3.73·7-s + 0.327·8-s + 6.26·9-s + 2.03·10-s − 2.44·11-s − 6.57·12-s + 2.67·13-s + 7.62·14-s − 3.04·15-s − 3.65·16-s − 4.31·17-s + 12.7·18-s + 4.90·19-s + 2.16·20-s − 11.3·21-s − 4.99·22-s + 3.82·23-s − 0.998·24-s + 25-s + 5.45·26-s − 9.93·27-s + 8.07·28-s + ⋯ |
L(s) = 1 | + 1.44·2-s − 1.75·3-s + 1.08·4-s + 0.447·5-s − 2.53·6-s + 1.41·7-s + 0.115·8-s + 2.08·9-s + 0.645·10-s − 0.738·11-s − 1.89·12-s + 0.742·13-s + 2.03·14-s − 0.785·15-s − 0.913·16-s − 1.04·17-s + 3.01·18-s + 1.12·19-s + 0.483·20-s − 2.48·21-s − 1.06·22-s + 0.796·23-s − 0.203·24-s + 0.200·25-s + 1.07·26-s − 1.91·27-s + 1.52·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.472362998\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.472362998\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 241 | \( 1 + T \) |
good | 2 | \( 1 - 2.03T + 2T^{2} \) |
| 3 | \( 1 + 3.04T + 3T^{2} \) |
| 7 | \( 1 - 3.73T + 7T^{2} \) |
| 11 | \( 1 + 2.44T + 11T^{2} \) |
| 13 | \( 1 - 2.67T + 13T^{2} \) |
| 17 | \( 1 + 4.31T + 17T^{2} \) |
| 19 | \( 1 - 4.90T + 19T^{2} \) |
| 23 | \( 1 - 3.82T + 23T^{2} \) |
| 29 | \( 1 + 1.38T + 29T^{2} \) |
| 31 | \( 1 - 7.03T + 31T^{2} \) |
| 37 | \( 1 - 9.99T + 37T^{2} \) |
| 41 | \( 1 - 0.473T + 41T^{2} \) |
| 43 | \( 1 - 6.70T + 43T^{2} \) |
| 47 | \( 1 - 4.83T + 47T^{2} \) |
| 53 | \( 1 + 4.16T + 53T^{2} \) |
| 59 | \( 1 - 9.97T + 59T^{2} \) |
| 61 | \( 1 + 0.618T + 61T^{2} \) |
| 67 | \( 1 - 12.8T + 67T^{2} \) |
| 71 | \( 1 + 12.6T + 71T^{2} \) |
| 73 | \( 1 + 10.7T + 73T^{2} \) |
| 79 | \( 1 + 6.32T + 79T^{2} \) |
| 83 | \( 1 - 9.56T + 83T^{2} \) |
| 89 | \( 1 + 11.7T + 89T^{2} \) |
| 97 | \( 1 - 6.00T + 97T^{2} \) |
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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
−10.19241667128388447608196118501, −8.989600302069824505468474739105, −7.74269318662563908775752260068, −6.79346136763231386108468177985, −5.97642771099085938378606133323, −5.40590478349778075929160891125, −4.79098843045854099060333708767, −4.19506933105725382380085691904, −2.55730638736389245116789820053, −1.12050867407271748511872204243,
1.12050867407271748511872204243, 2.55730638736389245116789820053, 4.19506933105725382380085691904, 4.79098843045854099060333708767, 5.40590478349778075929160891125, 5.97642771099085938378606133323, 6.79346136763231386108468177985, 7.74269318662563908775752260068, 8.989600302069824505468474739105, 10.19241667128388447608196118501