| L(s) = 1 | + 3-s − 4·7-s + 9-s + 4·11-s + 4·17-s + 4·19-s − 4·21-s − 4·23-s + 27-s − 2·29-s − 4·31-s + 4·33-s + 12·37-s + 12·41-s − 8·43-s + 9·49-s + 4·51-s − 14·53-s + 4·57-s − 2·59-s − 2·61-s − 4·63-s − 4·67-s − 4·69-s − 8·71-s − 6·73-s − 16·77-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.51·7-s + 1/3·9-s + 1.20·11-s + 0.970·17-s + 0.917·19-s − 0.872·21-s − 0.834·23-s + 0.192·27-s − 0.371·29-s − 0.718·31-s + 0.696·33-s + 1.97·37-s + 1.87·41-s − 1.21·43-s + 9/7·49-s + 0.560·51-s − 1.92·53-s + 0.529·57-s − 0.260·59-s − 0.256·61-s − 0.503·63-s − 0.488·67-s − 0.481·69-s − 0.949·71-s − 0.702·73-s − 1.82·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.578307447\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.578307447\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 167 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 12 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−13.21578631643929, −12.55446353464498, −12.19739127672469, −11.83485661465523, −11.16216701931302, −10.71166785131426, −9.982526252163350, −9.611772865755629, −9.323062214755505, −9.154373647522266, −8.181051337613743, −7.824014714760739, −7.397241414817767, −6.712757487089218, −6.380665935339169, −5.858281834871745, −5.438802604195094, −4.506484038097923, −4.015894944362860, −3.616179977341044, −2.980285457281289, −2.759291116401861, −1.739906106589661, −1.238704396505696, −0.4526849861750042,
0.4526849861750042, 1.238704396505696, 1.739906106589661, 2.759291116401861, 2.980285457281289, 3.616179977341044, 4.015894944362860, 4.506484038097923, 5.438802604195094, 5.858281834871745, 6.380665935339169, 6.712757487089218, 7.397241414817767, 7.824014714760739, 8.181051337613743, 9.154373647522266, 9.323062214755505, 9.611772865755629, 9.982526252163350, 10.71166785131426, 11.16216701931302, 11.83485661465523, 12.19739127672469, 12.55446353464498, 13.21578631643929
