| L(s) = 1 | − 3-s + 3.41·5-s + 1.41·7-s + 9-s + 4.82·11-s − 0.828·13-s − 3.41·15-s + 4.82·17-s − 2.82·19-s − 1.41·21-s − 1.17·23-s + 6.65·25-s − 27-s + 7.41·29-s − 7.07·31-s − 4.82·33-s + 4.82·35-s + 11.6·37-s + 0.828·39-s − 10.4·41-s − 6.82·43-s + 3.41·45-s − 12.4·47-s − 5·49-s − 4.82·51-s + 1.75·53-s + 16.4·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.52·5-s + 0.534·7-s + 0.333·9-s + 1.45·11-s − 0.229·13-s − 0.881·15-s + 1.17·17-s − 0.648·19-s − 0.308·21-s − 0.244·23-s + 1.33·25-s − 0.192·27-s + 1.37·29-s − 1.27·31-s − 0.840·33-s + 0.816·35-s + 1.91·37-s + 0.132·39-s − 1.63·41-s − 1.04·43-s + 0.508·45-s − 1.82·47-s − 0.714·49-s − 0.676·51-s + 0.241·53-s + 2.22·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.208226663\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.208226663\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| good | 5 | \( 1 - 3.41T + 5T^{2} \) |
| 7 | \( 1 - 1.41T + 7T^{2} \) |
| 11 | \( 1 - 4.82T + 11T^{2} \) |
| 13 | \( 1 + 0.828T + 13T^{2} \) |
| 17 | \( 1 - 4.82T + 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 + 1.17T + 23T^{2} \) |
| 29 | \( 1 - 7.41T + 29T^{2} \) |
| 31 | \( 1 + 7.07T + 31T^{2} \) |
| 37 | \( 1 - 11.6T + 37T^{2} \) |
| 41 | \( 1 + 10.4T + 41T^{2} \) |
| 43 | \( 1 + 6.82T + 43T^{2} \) |
| 47 | \( 1 + 12.4T + 47T^{2} \) |
| 53 | \( 1 - 1.75T + 53T^{2} \) |
| 59 | \( 1 + 1.65T + 59T^{2} \) |
| 61 | \( 1 - 0.343T + 61T^{2} \) |
| 67 | \( 1 - 5.65T + 67T^{2} \) |
| 71 | \( 1 - 8.48T + 71T^{2} \) |
| 73 | \( 1 + 11.3T + 73T^{2} \) |
| 79 | \( 1 - 17.4T + 79T^{2} \) |
| 83 | \( 1 - 8.82T + 83T^{2} \) |
| 89 | \( 1 - 5.31T + 89T^{2} \) |
| 97 | \( 1 + 7.65T + 97T^{2} \) |
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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
−9.747655769889656398421570804949, −8.788142312365327811825931283203, −7.889123220161755358339351386077, −6.60546057390317598997669417926, −6.33631621673338647429093132414, −5.36263466008609156728625107636, −4.68214789988357843469181011262, −3.44369405903479922806992837690, −2.00506838911489618005968151923, −1.23702908956003681583750718601,
1.23702908956003681583750718601, 2.00506838911489618005968151923, 3.44369405903479922806992837690, 4.68214789988357843469181011262, 5.36263466008609156728625107636, 6.33631621673338647429093132414, 6.60546057390317598997669417926, 7.889123220161755358339351386077, 8.788142312365327811825931283203, 9.747655769889656398421570804949
