L(s) = 1 | + 0.418·3-s + 0.0566·5-s − 7-s − 2.82·9-s + 11-s + 13-s + 0.0237·15-s + 1.81·17-s − 6.32·19-s − 0.418·21-s + 9.33·23-s − 4.99·25-s − 2.43·27-s + 2.91·29-s − 5.95·31-s + 0.418·33-s − 0.0566·35-s + 10.2·37-s + 0.418·39-s + 4.73·41-s − 4.59·43-s − 0.159·45-s + 9.33·47-s + 49-s + 0.760·51-s + 3.56·53-s + 0.0566·55-s + ⋯ |
L(s) = 1 | + 0.241·3-s + 0.0253·5-s − 0.377·7-s − 0.941·9-s + 0.301·11-s + 0.277·13-s + 0.00612·15-s + 0.440·17-s − 1.45·19-s − 0.0914·21-s + 1.94·23-s − 0.999·25-s − 0.469·27-s + 0.541·29-s − 1.06·31-s + 0.0729·33-s − 0.00957·35-s + 1.68·37-s + 0.0670·39-s + 0.739·41-s − 0.700·43-s − 0.0238·45-s + 1.36·47-s + 0.142·49-s + 0.106·51-s + 0.490·53-s + 0.00763·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.722102269\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.722102269\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
good | 3 | \( 1 - 0.418T + 3T^{2} \) |
| 5 | \( 1 - 0.0566T + 5T^{2} \) |
| 17 | \( 1 - 1.81T + 17T^{2} \) |
| 19 | \( 1 + 6.32T + 19T^{2} \) |
| 23 | \( 1 - 9.33T + 23T^{2} \) |
| 29 | \( 1 - 2.91T + 29T^{2} \) |
| 31 | \( 1 + 5.95T + 31T^{2} \) |
| 37 | \( 1 - 10.2T + 37T^{2} \) |
| 41 | \( 1 - 4.73T + 41T^{2} \) |
| 43 | \( 1 + 4.59T + 43T^{2} \) |
| 47 | \( 1 - 9.33T + 47T^{2} \) |
| 53 | \( 1 - 3.56T + 53T^{2} \) |
| 59 | \( 1 - 5.04T + 59T^{2} \) |
| 61 | \( 1 + 5.95T + 61T^{2} \) |
| 67 | \( 1 - 4.68T + 67T^{2} \) |
| 71 | \( 1 + 1.73T + 71T^{2} \) |
| 73 | \( 1 + 3.12T + 73T^{2} \) |
| 79 | \( 1 + 1.44T + 79T^{2} \) |
| 83 | \( 1 - 8.81T + 83T^{2} \) |
| 89 | \( 1 - 3.69T + 89T^{2} \) |
| 97 | \( 1 + 12.6T + 97T^{2} \) |
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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
−8.588770035932698002790931308648, −7.76245820482129611678212682167, −6.99207643829692806238549741510, −6.14637919533782851159461069219, −5.65350029132930006698890033283, −4.61557744050941878829766939041, −3.75837175677833723580434386272, −2.95369078131982193213310803701, −2.12386423202194109899587921368, −0.72946324586928882934103896415,
0.72946324586928882934103896415, 2.12386423202194109899587921368, 2.95369078131982193213310803701, 3.75837175677833723580434386272, 4.61557744050941878829766939041, 5.65350029132930006698890033283, 6.14637919533782851159461069219, 6.99207643829692806238549741510, 7.76245820482129611678212682167, 8.588770035932698002790931308648