| L(s) = 1 | + 3-s + 0.218·5-s + 4.47·7-s + 9-s − 3.58·11-s − 13-s + 0.218·15-s − 7.60·17-s + 4.65·19-s + 4.47·21-s + 5.21·23-s − 4.95·25-s + 27-s + 3.39·29-s − 1.71·31-s − 3.58·33-s + 0.978·35-s + 3.06·37-s − 39-s − 1.17·41-s + 6.53·43-s + 0.218·45-s + 6.69·47-s + 13.0·49-s − 7.60·51-s − 1.18·53-s − 0.782·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.0977·5-s + 1.69·7-s + 0.333·9-s − 1.08·11-s − 0.277·13-s + 0.0564·15-s − 1.84·17-s + 1.06·19-s + 0.977·21-s + 1.08·23-s − 0.990·25-s + 0.192·27-s + 0.629·29-s − 0.307·31-s − 0.623·33-s + 0.165·35-s + 0.504·37-s − 0.160·39-s − 0.183·41-s + 0.996·43-s + 0.0325·45-s + 0.976·47-s + 1.86·49-s − 1.06·51-s − 0.162·53-s − 0.105·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.107261004\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.107261004\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - 0.218T + 5T^{2} \) |
| 7 | \( 1 - 4.47T + 7T^{2} \) |
| 11 | \( 1 + 3.58T + 11T^{2} \) |
| 17 | \( 1 + 7.60T + 17T^{2} \) |
| 19 | \( 1 - 4.65T + 19T^{2} \) |
| 23 | \( 1 - 5.21T + 23T^{2} \) |
| 29 | \( 1 - 3.39T + 29T^{2} \) |
| 31 | \( 1 + 1.71T + 31T^{2} \) |
| 37 | \( 1 - 3.06T + 37T^{2} \) |
| 41 | \( 1 + 1.17T + 41T^{2} \) |
| 43 | \( 1 - 6.53T + 43T^{2} \) |
| 47 | \( 1 - 6.69T + 47T^{2} \) |
| 53 | \( 1 + 1.18T + 53T^{2} \) |
| 59 | \( 1 + 9.33T + 59T^{2} \) |
| 61 | \( 1 + 14.6T + 61T^{2} \) |
| 67 | \( 1 - 2.10T + 67T^{2} \) |
| 71 | \( 1 - 11.9T + 71T^{2} \) |
| 73 | \( 1 - 13.8T + 73T^{2} \) |
| 79 | \( 1 - 2.12T + 79T^{2} \) |
| 83 | \( 1 - 11.9T + 83T^{2} \) |
| 89 | \( 1 - 10.2T + 89T^{2} \) |
| 97 | \( 1 - 16.0T + 97T^{2} \) |
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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
−7.74405395227642644044659800130, −7.27407385005153623173760436294, −6.33109582416608555632258584525, −5.40495177626522922153759366079, −4.79822715616339073060882284285, −4.41911443073950858719677275127, −3.32053639463590452472493326709, −2.37583383939306889001930586729, −1.97072948690128540347551893407, −0.812811685759332509848298517500,
0.812811685759332509848298517500, 1.97072948690128540347551893407, 2.37583383939306889001930586729, 3.32053639463590452472493326709, 4.41911443073950858719677275127, 4.79822715616339073060882284285, 5.40495177626522922153759366079, 6.33109582416608555632258584525, 7.27407385005153623173760436294, 7.74405395227642644044659800130
