| L(s) = 1 | − 2-s + 3.31·3-s + 4-s − 1.82·5-s − 3.31·6-s − 8-s + 7.96·9-s + 1.82·10-s + 1.70·11-s + 3.31·12-s + 5.44·13-s − 6.04·15-s + 16-s − 4.05·17-s − 7.96·18-s − 5.79·19-s − 1.82·20-s − 1.70·22-s + 1.41·23-s − 3.31·24-s − 1.66·25-s − 5.44·26-s + 16.4·27-s + 4.04·29-s + 6.04·30-s + 3.70·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.91·3-s + 0.5·4-s − 0.816·5-s − 1.35·6-s − 0.353·8-s + 2.65·9-s + 0.577·10-s + 0.514·11-s + 0.955·12-s + 1.51·13-s − 1.56·15-s + 0.250·16-s − 0.983·17-s − 1.87·18-s − 1.32·19-s − 0.408·20-s − 0.363·22-s + 0.295·23-s − 0.675·24-s − 0.333·25-s − 1.06·26-s + 3.16·27-s + 0.751·29-s + 1.10·30-s + 0.665·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7742 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7742 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.031308976\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.031308976\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| 79 | \( 1 + T \) |
| good | 3 | \( 1 - 3.31T + 3T^{2} \) |
| 5 | \( 1 + 1.82T + 5T^{2} \) |
| 11 | \( 1 - 1.70T + 11T^{2} \) |
| 13 | \( 1 - 5.44T + 13T^{2} \) |
| 17 | \( 1 + 4.05T + 17T^{2} \) |
| 19 | \( 1 + 5.79T + 19T^{2} \) |
| 23 | \( 1 - 1.41T + 23T^{2} \) |
| 29 | \( 1 - 4.04T + 29T^{2} \) |
| 31 | \( 1 - 3.70T + 31T^{2} \) |
| 37 | \( 1 + 3.82T + 37T^{2} \) |
| 41 | \( 1 - 6.73T + 41T^{2} \) |
| 43 | \( 1 - 8.64T + 43T^{2} \) |
| 47 | \( 1 + 1.65T + 47T^{2} \) |
| 53 | \( 1 + 10.0T + 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 + 2.70T + 61T^{2} \) |
| 67 | \( 1 - 3.44T + 67T^{2} \) |
| 71 | \( 1 - 5.57T + 71T^{2} \) |
| 73 | \( 1 + 1.63T + 73T^{2} \) |
| 83 | \( 1 - 14.7T + 83T^{2} \) |
| 89 | \( 1 - 4.93T + 89T^{2} \) |
| 97 | \( 1 - 12.3T + 97T^{2} \) |
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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
−8.110236390289905140676866314177, −7.50959407071813224111934999982, −6.66199760316196026902309777487, −6.26751895391774855490911141509, −4.61034503928625148420416236695, −3.98538144821232835806954875446, −3.52314981974828566245257575827, −2.59072343853073019389282348884, −1.89437276341179655613822148171, −0.909066636571351018660185731821,
0.909066636571351018660185731821, 1.89437276341179655613822148171, 2.59072343853073019389282348884, 3.52314981974828566245257575827, 3.98538144821232835806954875446, 4.61034503928625148420416236695, 6.26751895391774855490911141509, 6.66199760316196026902309777487, 7.50959407071813224111934999982, 8.110236390289905140676866314177
