| L(s) = 1 | + 3-s + 2·7-s + 9-s − 4.47·11-s − 4.47·13-s + 4.47·17-s + 2·21-s + 4·23-s + 27-s + 4·29-s + 8.94·31-s − 4.47·33-s + 4.47·37-s − 4.47·39-s + 10·41-s − 4·43-s + 8·47-s − 3·49-s + 4.47·51-s + 4.47·53-s + 13.4·59-s + 10·61-s + 2·63-s − 8·67-s + 4·69-s − 8.94·71-s + 8.94·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.755·7-s + 0.333·9-s − 1.34·11-s − 1.24·13-s + 1.08·17-s + 0.436·21-s + 0.834·23-s + 0.192·27-s + 0.742·29-s + 1.60·31-s − 0.778·33-s + 0.735·37-s − 0.716·39-s + 1.56·41-s − 0.609·43-s + 1.16·47-s − 0.428·49-s + 0.626·51-s + 0.614·53-s + 1.74·59-s + 1.28·61-s + 0.251·63-s − 0.977·67-s + 0.481·69-s − 1.06·71-s + 1.04·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.297841355\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.297841355\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 + 4.47T + 11T^{2} \) |
| 13 | \( 1 + 4.47T + 13T^{2} \) |
| 17 | \( 1 - 4.47T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 - 8.94T + 31T^{2} \) |
| 37 | \( 1 - 4.47T + 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 - 4.47T + 53T^{2} \) |
| 59 | \( 1 - 13.4T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 + 8.94T + 71T^{2} \) |
| 73 | \( 1 - 8.94T + 73T^{2} \) |
| 79 | \( 1 + 8.94T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 17.8T + 97T^{2} \) |
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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
−8.828228639670930280770495358102, −8.050894721563563897044808796100, −7.66209956965641345430393745894, −6.88689601251649846872269261400, −5.62347779434008909036818340706, −4.98876645031598159722284006617, −4.24537533360346946289565811614, −2.87151009502897082228150529747, −2.44515175279066703071556712507, −0.969589636468131946357116609884,
0.969589636468131946357116609884, 2.44515175279066703071556712507, 2.87151009502897082228150529747, 4.24537533360346946289565811614, 4.98876645031598159722284006617, 5.62347779434008909036818340706, 6.88689601251649846872269261400, 7.66209956965641345430393745894, 8.050894721563563897044808796100, 8.828228639670930280770495358102
