| L(s) = 1 | − 1.61·2-s + 2.61·3-s + 0.618·4-s − 4.23·6-s + 2.85·7-s + 2.23·8-s + 3.85·9-s + 1.61·12-s − 6.23·13-s − 4.61·14-s − 4.85·16-s + 0.618·17-s − 6.23·18-s + 6.70·19-s + 7.47·21-s − 4.09·23-s + 5.85·24-s + 10.0·26-s + 2.23·27-s + 1.76·28-s + 1.38·29-s − 3·31-s + 3.38·32-s − 1.00·34-s + 2.38·36-s + 10.2·37-s − 10.8·38-s + ⋯ |
| L(s) = 1 | − 1.14·2-s + 1.51·3-s + 0.309·4-s − 1.72·6-s + 1.07·7-s + 0.790·8-s + 1.28·9-s + 0.467·12-s − 1.72·13-s − 1.23·14-s − 1.21·16-s + 0.149·17-s − 1.46·18-s + 1.53·19-s + 1.63·21-s − 0.852·23-s + 1.19·24-s + 1.97·26-s + 0.430·27-s + 0.333·28-s + 0.256·29-s − 0.538·31-s + 0.597·32-s − 0.171·34-s + 0.396·36-s + 1.68·37-s − 1.76·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.860039899\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.860039899\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 1.61T + 2T^{2} \) |
| 3 | \( 1 - 2.61T + 3T^{2} \) |
| 7 | \( 1 - 2.85T + 7T^{2} \) |
| 13 | \( 1 + 6.23T + 13T^{2} \) |
| 17 | \( 1 - 0.618T + 17T^{2} \) |
| 19 | \( 1 - 6.70T + 19T^{2} \) |
| 23 | \( 1 + 4.09T + 23T^{2} \) |
| 29 | \( 1 - 1.38T + 29T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 - 10.2T + 37T^{2} \) |
| 41 | \( 1 - 3T + 41T^{2} \) |
| 43 | \( 1 - 6T + 43T^{2} \) |
| 47 | \( 1 - 11.9T + 47T^{2} \) |
| 53 | \( 1 - 9.32T + 53T^{2} \) |
| 59 | \( 1 - 0.527T + 59T^{2} \) |
| 61 | \( 1 + 0.0901T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 - 8.18T + 71T^{2} \) |
| 73 | \( 1 + 10.3T + 73T^{2} \) |
| 79 | \( 1 + 5.85T + 79T^{2} \) |
| 83 | \( 1 - 10.1T + 83T^{2} \) |
| 89 | \( 1 + 6.90T + 89T^{2} \) |
| 97 | \( 1 - 1.61T + 97T^{2} \) |
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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
−8.723182933114141774356233014989, −7.982127841784803510951837089652, −7.55963668110719572475454539421, −7.23302963253273772554584265667, −5.57564005844318988557181258662, −4.67646846008104447760636381898, −3.95106545787954939627133487164, −2.64681714330060234571015218646, −2.09291427244085619948591458458, −0.956431036243189684222132538883,
0.956431036243189684222132538883, 2.09291427244085619948591458458, 2.64681714330060234571015218646, 3.95106545787954939627133487164, 4.67646846008104447760636381898, 5.57564005844318988557181258662, 7.23302963253273772554584265667, 7.55963668110719572475454539421, 7.982127841784803510951837089652, 8.723182933114141774356233014989
