| L(s) = 1 | + 3.13·3-s − 0.555·5-s − 7-s + 6.81·9-s + 3.14·11-s − 4.47·13-s − 1.74·15-s − 0.980·17-s + 7.44·19-s − 3.13·21-s + 1.25·23-s − 4.69·25-s + 11.9·27-s − 4.48·29-s + 4.43·31-s + 9.86·33-s + 0.555·35-s + 0.912·37-s − 14.0·39-s + 1.21·41-s + 1.36·43-s − 3.79·45-s + 9.97·47-s + 49-s − 3.07·51-s + 11.4·53-s − 1.75·55-s + ⋯ |
| L(s) = 1 | + 1.80·3-s − 0.248·5-s − 0.377·7-s + 2.27·9-s + 0.949·11-s − 1.24·13-s − 0.449·15-s − 0.237·17-s + 1.70·19-s − 0.683·21-s + 0.262·23-s − 0.938·25-s + 2.30·27-s − 0.833·29-s + 0.795·31-s + 1.71·33-s + 0.0939·35-s + 0.149·37-s − 2.24·39-s + 0.189·41-s + 0.208·43-s − 0.565·45-s + 1.45·47-s + 0.142·49-s − 0.430·51-s + 1.56·53-s − 0.235·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.113318091\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.113318091\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 - 3.13T + 3T^{2} \) |
| 5 | \( 1 + 0.555T + 5T^{2} \) |
| 11 | \( 1 - 3.14T + 11T^{2} \) |
| 13 | \( 1 + 4.47T + 13T^{2} \) |
| 17 | \( 1 + 0.980T + 17T^{2} \) |
| 19 | \( 1 - 7.44T + 19T^{2} \) |
| 23 | \( 1 - 1.25T + 23T^{2} \) |
| 29 | \( 1 + 4.48T + 29T^{2} \) |
| 31 | \( 1 - 4.43T + 31T^{2} \) |
| 37 | \( 1 - 0.912T + 37T^{2} \) |
| 41 | \( 1 - 1.21T + 41T^{2} \) |
| 43 | \( 1 - 1.36T + 43T^{2} \) |
| 47 | \( 1 - 9.97T + 47T^{2} \) |
| 53 | \( 1 - 11.4T + 53T^{2} \) |
| 59 | \( 1 + 2.56T + 59T^{2} \) |
| 61 | \( 1 - 3.65T + 61T^{2} \) |
| 67 | \( 1 + 2.25T + 67T^{2} \) |
| 71 | \( 1 - 0.934T + 71T^{2} \) |
| 73 | \( 1 - 0.710T + 73T^{2} \) |
| 79 | \( 1 - 11.8T + 79T^{2} \) |
| 83 | \( 1 - 9.58T + 83T^{2} \) |
| 89 | \( 1 + 10.2T + 89T^{2} \) |
| 97 | \( 1 + 3.03T + 97T^{2} \) |
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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
−7.83665328000154189839453548943, −7.37217445016174782486645892548, −6.95866111065162492356592139873, −5.86709582811229946184961123307, −4.87108296789916262374487677939, −4.02009773531741536952431650441, −3.53968244216373175908659291815, −2.72517543337496059707978576375, −2.09167673774931658071232990726, −0.968529273152825304684811666735,
0.968529273152825304684811666735, 2.09167673774931658071232990726, 2.72517543337496059707978576375, 3.53968244216373175908659291815, 4.02009773531741536952431650441, 4.87108296789916262374487677939, 5.86709582811229946184961123307, 6.95866111065162492356592139873, 7.37217445016174782486645892548, 7.83665328000154189839453548943
