L(s) = 1 | + 0.717·2-s − 0.928·3-s − 1.48·4-s − 0.666·6-s + 1.52·7-s − 2.50·8-s − 2.13·9-s + 4.86·11-s + 1.37·12-s + 3.32·13-s + 1.09·14-s + 1.17·16-s + 8.03·17-s − 1.53·18-s + 0.900·19-s − 1.41·21-s + 3.49·22-s − 2.61·23-s + 2.32·24-s + 2.38·26-s + 4.76·27-s − 2.26·28-s − 5.55·29-s + 0.512·31-s + 5.84·32-s − 4.51·33-s + 5.76·34-s + ⋯ |
L(s) = 1 | + 0.507·2-s − 0.535·3-s − 0.742·4-s − 0.272·6-s + 0.576·7-s − 0.884·8-s − 0.712·9-s + 1.46·11-s + 0.397·12-s + 0.921·13-s + 0.292·14-s + 0.293·16-s + 1.94·17-s − 0.361·18-s + 0.206·19-s − 0.308·21-s + 0.744·22-s − 0.544·23-s + 0.473·24-s + 0.467·26-s + 0.917·27-s − 0.427·28-s − 1.03·29-s + 0.0920·31-s + 1.03·32-s − 0.786·33-s + 0.988·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.954019156\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.954019156\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
good | 2 | \( 1 - 0.717T + 2T^{2} \) |
| 3 | \( 1 + 0.928T + 3T^{2} \) |
| 7 | \( 1 - 1.52T + 7T^{2} \) |
| 11 | \( 1 - 4.86T + 11T^{2} \) |
| 13 | \( 1 - 3.32T + 13T^{2} \) |
| 17 | \( 1 - 8.03T + 17T^{2} \) |
| 19 | \( 1 - 0.900T + 19T^{2} \) |
| 23 | \( 1 + 2.61T + 23T^{2} \) |
| 29 | \( 1 + 5.55T + 29T^{2} \) |
| 31 | \( 1 - 0.512T + 31T^{2} \) |
| 37 | \( 1 + 6.28T + 37T^{2} \) |
| 41 | \( 1 + 0.544T + 41T^{2} \) |
| 43 | \( 1 - 2.92T + 43T^{2} \) |
| 47 | \( 1 - 3.68T + 47T^{2} \) |
| 53 | \( 1 - 8.48T + 53T^{2} \) |
| 59 | \( 1 + 4.89T + 59T^{2} \) |
| 61 | \( 1 + 8.63T + 61T^{2} \) |
| 67 | \( 1 + 6.25T + 67T^{2} \) |
| 71 | \( 1 + 12.6T + 71T^{2} \) |
| 73 | \( 1 - 9.81T + 73T^{2} \) |
| 79 | \( 1 - 12.8T + 79T^{2} \) |
| 83 | \( 1 + 8.17T + 83T^{2} \) |
| 89 | \( 1 + 3.31T + 89T^{2} \) |
| 97 | \( 1 - 13.9T + 97T^{2} \) |
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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
−8.135126820863232073035775197330, −7.39300513125446585887819440429, −6.27663387638654730076496577119, −5.84287664722932981457477275741, −5.31369480493312420027464587627, −4.47069384254764580548974165020, −3.65672359375855363310561009092, −3.22319149794565069827812470162, −1.62806701258419936908133302597, −0.75594987391056671760733988397,
0.75594987391056671760733988397, 1.62806701258419936908133302597, 3.22319149794565069827812470162, 3.65672359375855363310561009092, 4.47069384254764580548974165020, 5.31369480493312420027464587627, 5.84287664722932981457477275741, 6.27663387638654730076496577119, 7.39300513125446585887819440429, 8.135126820863232073035775197330