L(s) = 1 | − 2.64·3-s − 1.73i·5-s + 4.00·9-s − 4.58i·11-s + 3.46i·13-s + 4.58i·15-s + 5.19i·17-s − 2.64·19-s − 4.58i·23-s + 2.00·25-s − 2.64·27-s − 2.64·31-s + 12.1i·33-s − 7·37-s − 9.16i·39-s + ⋯ |
L(s) = 1 | − 1.52·3-s − 0.774i·5-s + 1.33·9-s − 1.38i·11-s + 0.960i·13-s + 1.18i·15-s + 1.26i·17-s − 0.606·19-s − 0.955i·23-s + 0.400·25-s − 0.509·27-s − 0.475·31-s + 2.11i·33-s − 1.15·37-s − 1.46i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6413188053\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6413188053\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + 2.64T + 3T^{2} \) |
| 5 | \( 1 + 1.73iT - 5T^{2} \) |
| 11 | \( 1 + 4.58iT - 11T^{2} \) |
| 13 | \( 1 - 3.46iT - 13T^{2} \) |
| 17 | \( 1 - 5.19iT - 17T^{2} \) |
| 19 | \( 1 + 2.64T + 19T^{2} \) |
| 23 | \( 1 + 4.58iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 2.64T + 31T^{2} \) |
| 37 | \( 1 + 7T + 37T^{2} \) |
| 41 | \( 1 - 3.46iT - 41T^{2} \) |
| 43 | \( 1 - 9.16iT - 43T^{2} \) |
| 47 | \( 1 + 7.93T + 47T^{2} \) |
| 53 | \( 1 + 3T + 53T^{2} \) |
| 59 | \( 1 - 7.93T + 59T^{2} \) |
| 61 | \( 1 - 1.73iT - 61T^{2} \) |
| 67 | \( 1 - 4.58iT - 67T^{2} \) |
| 71 | \( 1 + 9.16iT - 71T^{2} \) |
| 73 | \( 1 + 5.19iT - 73T^{2} \) |
| 79 | \( 1 + 4.58iT - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 1.73iT - 89T^{2} \) |
| 97 | \( 1 - 3.46iT - 97T^{2} \) |
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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
−8.603959685117605274624274799275, −8.281390907206189276589714129924, −6.95873441802458218855721305703, −6.26425990786733872315462468550, −5.89381048525272325134644529847, −4.92099515484713832889273029018, −4.42283027754529808553492740985, −3.37918003264812349990272047166, −1.78507192671630068897529129916, −0.78695305702569501171158052764,
0.35230119539544525332270678223, 1.79966140211729583526849501064, 2.94929372254677693193953866936, 4.03869918000130761606394770362, 5.14314880454126610952100331277, 5.32539454172096132771650614404, 6.41825661967555690090194121871, 7.08363225196803871112505006140, 7.38198533927081371147750091926, 8.593323387202409771864500654559