| L(s) = 1 | + 2.75·2-s + 1.49·3-s + 5.60·4-s + 4.11·6-s + 2.84·7-s + 9.94·8-s − 0.774·9-s − 0.864·11-s + 8.36·12-s + 0.643·13-s + 7.85·14-s + 16.2·16-s − 3.74·17-s − 2.13·18-s + 4.24·21-s − 2.38·22-s − 0.417·23-s + 14.8·24-s + 1.77·26-s − 5.63·27-s + 15.9·28-s + 9.70·29-s − 4.93·31-s + 24.8·32-s − 1.29·33-s − 10.3·34-s − 4.34·36-s + ⋯ |
| L(s) = 1 | + 1.95·2-s + 0.861·3-s + 2.80·4-s + 1.67·6-s + 1.07·7-s + 3.51·8-s − 0.258·9-s − 0.260·11-s + 2.41·12-s + 0.178·13-s + 2.09·14-s + 4.05·16-s − 0.907·17-s − 0.503·18-s + 0.927·21-s − 0.508·22-s − 0.0870·23-s + 3.02·24-s + 0.347·26-s − 1.08·27-s + 3.01·28-s + 1.80·29-s − 0.886·31-s + 4.39·32-s − 0.224·33-s − 1.76·34-s − 0.723·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(12.31385495\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.31385495\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 2.75T + 2T^{2} \) |
| 3 | \( 1 - 1.49T + 3T^{2} \) |
| 7 | \( 1 - 2.84T + 7T^{2} \) |
| 11 | \( 1 + 0.864T + 11T^{2} \) |
| 13 | \( 1 - 0.643T + 13T^{2} \) |
| 17 | \( 1 + 3.74T + 17T^{2} \) |
| 23 | \( 1 + 0.417T + 23T^{2} \) |
| 29 | \( 1 - 9.70T + 29T^{2} \) |
| 31 | \( 1 + 4.93T + 31T^{2} \) |
| 37 | \( 1 - 6.36T + 37T^{2} \) |
| 41 | \( 1 - 4.01T + 41T^{2} \) |
| 43 | \( 1 - 2.05T + 43T^{2} \) |
| 47 | \( 1 - 3.95T + 47T^{2} \) |
| 53 | \( 1 + 10.9T + 53T^{2} \) |
| 59 | \( 1 + 2.45T + 59T^{2} \) |
| 61 | \( 1 - 6.33T + 61T^{2} \) |
| 67 | \( 1 + 2.53T + 67T^{2} \) |
| 71 | \( 1 - 1.78T + 71T^{2} \) |
| 73 | \( 1 - 7.13T + 73T^{2} \) |
| 79 | \( 1 + 1.82T + 79T^{2} \) |
| 83 | \( 1 - 7.43T + 83T^{2} \) |
| 89 | \( 1 + 4.44T + 89T^{2} \) |
| 97 | \( 1 + 10.8T + 97T^{2} \) |
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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
−7.73286900395089884205998342392, −6.84976634337694986141746734805, −6.19878433548996243280833430672, −5.49619519517840036477797165948, −4.79646293763180028201366383522, −4.29229782423517606501374844596, −3.56275660414593274498909694785, −2.68683150592785329526508798278, −2.31109042727214224982455867084, −1.37072949960239552813855742296,
1.37072949960239552813855742296, 2.31109042727214224982455867084, 2.68683150592785329526508798278, 3.56275660414593274498909694785, 4.29229782423517606501374844596, 4.79646293763180028201366383522, 5.49619519517840036477797165948, 6.19878433548996243280833430672, 6.84976634337694986141746734805, 7.73286900395089884205998342392
