| L(s) = 1 | + 0.328·2-s − 0.962·3-s − 1.89·4-s − 0.315·6-s − 3.27·7-s − 1.27·8-s − 2.07·9-s + 1.82·12-s − 5.20·13-s − 1.07·14-s + 3.36·16-s − 3.60·17-s − 0.681·18-s − 5.00·19-s + 3.14·21-s − 5.84·23-s + 1.22·24-s − 1.70·26-s + 4.88·27-s + 6.19·28-s + 2.04·29-s − 3.25·31-s + 3.66·32-s − 1.18·34-s + 3.92·36-s − 6.23·37-s − 1.64·38-s + ⋯ |
| L(s) = 1 | + 0.232·2-s − 0.555·3-s − 0.946·4-s − 0.128·6-s − 1.23·7-s − 0.451·8-s − 0.691·9-s + 0.525·12-s − 1.44·13-s − 0.287·14-s + 0.841·16-s − 0.873·17-s − 0.160·18-s − 1.14·19-s + 0.686·21-s − 1.21·23-s + 0.250·24-s − 0.335·26-s + 0.939·27-s + 1.17·28-s + 0.380·29-s − 0.584·31-s + 0.647·32-s − 0.202·34-s + 0.654·36-s − 1.02·37-s − 0.266·38-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
| \(L(1)\) |
\(\approx\) |
\(0.05308424562\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05308424562\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.328T + 2T^{2} \) |
| 3 | \( 1 + 0.962T + 3T^{2} \) |
| 7 | \( 1 + 3.27T + 7T^{2} \) |
| 13 | \( 1 + 5.20T + 13T^{2} \) |
| 17 | \( 1 + 3.60T + 17T^{2} \) |
| 19 | \( 1 + 5.00T + 19T^{2} \) |
| 23 | \( 1 + 5.84T + 23T^{2} \) |
| 29 | \( 1 - 2.04T + 29T^{2} \) |
| 31 | \( 1 + 3.25T + 31T^{2} \) |
| 37 | \( 1 + 6.23T + 37T^{2} \) |
| 41 | \( 1 + 10.2T + 41T^{2} \) |
| 43 | \( 1 + 0.596T + 43T^{2} \) |
| 47 | \( 1 - 0.962T + 47T^{2} \) |
| 53 | \( 1 + 0.393T + 53T^{2} \) |
| 59 | \( 1 + 0.527T + 59T^{2} \) |
| 61 | \( 1 - 5.32T + 61T^{2} \) |
| 67 | \( 1 + 8.45T + 67T^{2} \) |
| 71 | \( 1 - 12.5T + 71T^{2} \) |
| 73 | \( 1 - 8.51T + 73T^{2} \) |
| 79 | \( 1 + 2.83T + 79T^{2} \) |
| 83 | \( 1 + 12.1T + 83T^{2} \) |
| 89 | \( 1 - 9T + 89T^{2} \) |
| 97 | \( 1 + 11.2T + 97T^{2} \) |
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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
−8.738281669188726968501986330577, −8.170888529026599273743921609646, −6.94717037544381118232642152131, −6.40573513340664575325318967987, −5.59615965177099585213765375493, −4.91488550208582936438923486575, −4.09287198010311237144947920566, −3.20907061878371442989783727767, −2.22511130503367612726236259510, −0.12896508228301528211508353972,
0.12896508228301528211508353972, 2.22511130503367612726236259510, 3.20907061878371442989783727767, 4.09287198010311237144947920566, 4.91488550208582936438923486575, 5.59615965177099585213765375493, 6.40573513340664575325318967987, 6.94717037544381118232642152131, 8.170888529026599273743921609646, 8.738281669188726968501986330577
