| L(s) = 1 | + 2-s − 3.13·3-s + 4-s + 3.00·5-s − 3.13·6-s − 0.324·7-s + 8-s + 6.84·9-s + 3.00·10-s − 4.08·11-s − 3.13·12-s − 0.304·13-s − 0.324·14-s − 9.43·15-s + 16-s + 5.91·17-s + 6.84·18-s + 2.22·19-s + 3.00·20-s + 1.01·21-s − 4.08·22-s + 8.27·23-s − 3.13·24-s + 4.03·25-s − 0.304·26-s − 12.0·27-s − 0.324·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.81·3-s + 0.5·4-s + 1.34·5-s − 1.28·6-s − 0.122·7-s + 0.353·8-s + 2.28·9-s + 0.950·10-s − 1.23·11-s − 0.905·12-s − 0.0843·13-s − 0.0866·14-s − 2.43·15-s + 0.250·16-s + 1.43·17-s + 1.61·18-s + 0.510·19-s + 0.672·20-s + 0.222·21-s − 0.870·22-s + 1.72·23-s − 0.640·24-s + 0.806·25-s − 0.0596·26-s − 2.32·27-s − 0.0613·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.575477346\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.575477346\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 269 | \( 1 + T \) |
| good | 3 | \( 1 + 3.13T + 3T^{2} \) |
| 5 | \( 1 - 3.00T + 5T^{2} \) |
| 7 | \( 1 + 0.324T + 7T^{2} \) |
| 11 | \( 1 + 4.08T + 11T^{2} \) |
| 13 | \( 1 + 0.304T + 13T^{2} \) |
| 17 | \( 1 - 5.91T + 17T^{2} \) |
| 19 | \( 1 - 2.22T + 19T^{2} \) |
| 23 | \( 1 - 8.27T + 23T^{2} \) |
| 29 | \( 1 + 5.84T + 29T^{2} \) |
| 31 | \( 1 - 10.2T + 31T^{2} \) |
| 37 | \( 1 - 3.68T + 37T^{2} \) |
| 41 | \( 1 - 5.70T + 41T^{2} \) |
| 43 | \( 1 - 3.80T + 43T^{2} \) |
| 47 | \( 1 + 10.2T + 47T^{2} \) |
| 53 | \( 1 - 6.39T + 53T^{2} \) |
| 59 | \( 1 + 2.31T + 59T^{2} \) |
| 61 | \( 1 + 8.06T + 61T^{2} \) |
| 67 | \( 1 - 7.93T + 67T^{2} \) |
| 71 | \( 1 - 7.43T + 71T^{2} \) |
| 73 | \( 1 + 16.5T + 73T^{2} \) |
| 79 | \( 1 + 1.09T + 79T^{2} \) |
| 83 | \( 1 + 5.59T + 83T^{2} \) |
| 89 | \( 1 + 14.9T + 89T^{2} \) |
| 97 | \( 1 + 2.53T + 97T^{2} \) |
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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
−10.93137068211047070504514086750, −10.13430081778984582895193967991, −9.631758177173416415477140304270, −7.73222241289680362605640884011, −6.77121060459657425456851312627, −5.84070545962247601371267929583, −5.43184937882695645980290119454, −4.69853718509495514457637834160, −2.87572279662339847900962907072, −1.22534948810573436086387148245,
1.22534948810573436086387148245, 2.87572279662339847900962907072, 4.69853718509495514457637834160, 5.43184937882695645980290119454, 5.84070545962247601371267929583, 6.77121060459657425456851312627, 7.73222241289680362605640884011, 9.631758177173416415477140304270, 10.13430081778984582895193967991, 10.93137068211047070504514086750
