L(s) = 1 | + 2.07·2-s − 3.01·3-s + 2.29·4-s + 2.69·5-s − 6.24·6-s − 7-s + 0.602·8-s + 6.09·9-s + 5.57·10-s − 1.66·11-s − 6.90·12-s − 2.07·14-s − 8.12·15-s − 3.33·16-s + 6.90·17-s + 12.6·18-s + 7.92·19-s + 6.16·20-s + 3.01·21-s − 3.43·22-s + 1.95·23-s − 1.81·24-s + 2.24·25-s − 9.34·27-s − 2.29·28-s − 2.71·29-s − 16.8·30-s + ⋯ |
L(s) = 1 | + 1.46·2-s − 1.74·3-s + 1.14·4-s + 1.20·5-s − 2.55·6-s − 0.377·7-s + 0.212·8-s + 2.03·9-s + 1.76·10-s − 0.500·11-s − 1.99·12-s − 0.553·14-s − 2.09·15-s − 0.833·16-s + 1.67·17-s + 2.97·18-s + 1.81·19-s + 1.37·20-s + 0.658·21-s − 0.733·22-s + 0.406·23-s − 0.370·24-s + 0.449·25-s − 1.79·27-s − 0.432·28-s − 0.504·29-s − 3.07·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.466916365\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.466916365\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 2.07T + 2T^{2} \) |
| 3 | \( 1 + 3.01T + 3T^{2} \) |
| 5 | \( 1 - 2.69T + 5T^{2} \) |
| 11 | \( 1 + 1.66T + 11T^{2} \) |
| 17 | \( 1 - 6.90T + 17T^{2} \) |
| 19 | \( 1 - 7.92T + 19T^{2} \) |
| 23 | \( 1 - 1.95T + 23T^{2} \) |
| 29 | \( 1 + 2.71T + 29T^{2} \) |
| 31 | \( 1 - 1.76T + 31T^{2} \) |
| 37 | \( 1 - 4.20T + 37T^{2} \) |
| 41 | \( 1 - 7.08T + 41T^{2} \) |
| 43 | \( 1 + 3.56T + 43T^{2} \) |
| 47 | \( 1 - 1.53T + 47T^{2} \) |
| 53 | \( 1 - 12.7T + 53T^{2} \) |
| 59 | \( 1 + 4.52T + 59T^{2} \) |
| 61 | \( 1 - 2.89T + 61T^{2} \) |
| 67 | \( 1 - 1.27T + 67T^{2} \) |
| 71 | \( 1 - 2.93T + 71T^{2} \) |
| 73 | \( 1 - 14.6T + 73T^{2} \) |
| 79 | \( 1 - 2.91T + 79T^{2} \) |
| 83 | \( 1 - 2.33T + 83T^{2} \) |
| 89 | \( 1 + 7.64T + 89T^{2} \) |
| 97 | \( 1 + 12.9T + 97T^{2} \) |
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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
−9.900336344068608642848866055211, −9.490547203013633365476452846330, −7.61281213254363776617537825295, −6.74588386985525752114477528799, −5.93759151162925812035093571507, −5.41414956814245708342459139793, −5.16470703608337657912324105949, −3.85522500894432201053823130640, −2.71016511190279063583640429759, −1.10496171521776825972041254786,
1.10496171521776825972041254786, 2.71016511190279063583640429759, 3.85522500894432201053823130640, 5.16470703608337657912324105949, 5.41414956814245708342459139793, 5.93759151162925812035093571507, 6.74588386985525752114477528799, 7.61281213254363776617537825295, 9.490547203013633365476452846330, 9.900336344068608642848866055211