| L(s) = 1 | − 1.56·5-s + 3·7-s + 3.56·11-s + 2.56·13-s + 8.12·17-s + 19-s − 1.43·23-s − 2.56·25-s + 7.68·29-s + 0.876·31-s − 4.68·35-s − 1.12·37-s − 4·41-s + 9.56·43-s − 8.68·47-s + 2·49-s + 8.56·53-s − 5.56·55-s + 8.56·59-s − 5.80·61-s − 4·65-s − 4.56·67-s − 12.2·71-s + 7.24·73-s + 10.6·77-s + 10·79-s − 7.36·83-s + ⋯ |
| L(s) = 1 | − 0.698·5-s + 1.13·7-s + 1.07·11-s + 0.710·13-s + 1.97·17-s + 0.229·19-s − 0.299·23-s − 0.512·25-s + 1.42·29-s + 0.157·31-s − 0.791·35-s − 0.184·37-s − 0.624·41-s + 1.45·43-s − 1.26·47-s + 0.285·49-s + 1.17·53-s − 0.749·55-s + 1.11·59-s − 0.743·61-s − 0.496·65-s − 0.557·67-s − 1.45·71-s + 0.848·73-s + 1.21·77-s + 1.12·79-s − 0.808·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5472 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.525140013\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.525140013\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + 1.56T + 5T^{2} \) |
| 7 | \( 1 - 3T + 7T^{2} \) |
| 11 | \( 1 - 3.56T + 11T^{2} \) |
| 13 | \( 1 - 2.56T + 13T^{2} \) |
| 17 | \( 1 - 8.12T + 17T^{2} \) |
| 23 | \( 1 + 1.43T + 23T^{2} \) |
| 29 | \( 1 - 7.68T + 29T^{2} \) |
| 31 | \( 1 - 0.876T + 31T^{2} \) |
| 37 | \( 1 + 1.12T + 37T^{2} \) |
| 41 | \( 1 + 4T + 41T^{2} \) |
| 43 | \( 1 - 9.56T + 43T^{2} \) |
| 47 | \( 1 + 8.68T + 47T^{2} \) |
| 53 | \( 1 - 8.56T + 53T^{2} \) |
| 59 | \( 1 - 8.56T + 59T^{2} \) |
| 61 | \( 1 + 5.80T + 61T^{2} \) |
| 67 | \( 1 + 4.56T + 67T^{2} \) |
| 71 | \( 1 + 12.2T + 71T^{2} \) |
| 73 | \( 1 - 7.24T + 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 + 7.36T + 83T^{2} \) |
| 89 | \( 1 + 9.36T + 89T^{2} \) |
| 97 | \( 1 + 1.12T + 97T^{2} \) |
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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
−8.172817810352951082318929504919, −7.60684804537190771448330602739, −6.82532182922713467908780470717, −5.94312904475062751970603614611, −5.28049122798695779775084390241, −4.35730989303304622860553620190, −3.80242894078641353783412360705, −2.97834202402391071147252240709, −1.60667607946198771711868199936, −0.961262508287906583155104181122,
0.961262508287906583155104181122, 1.60667607946198771711868199936, 2.97834202402391071147252240709, 3.80242894078641353783412360705, 4.35730989303304622860553620190, 5.28049122798695779775084390241, 5.94312904475062751970603614611, 6.82532182922713467908780470717, 7.60684804537190771448330602739, 8.172817810352951082318929504919
