L(s) = 1 | − 3.13·3-s + 5-s + 1.68·7-s + 6.81·9-s + 0.548·13-s − 3.13·15-s + 4.26·17-s + 6.81·19-s − 5.26·21-s + 2.81·23-s + 25-s − 11.9·27-s + 10.2·29-s − 7.71·31-s + 1.68·35-s + 9.36·37-s − 1.71·39-s + 10.0·41-s + 5.94·43-s + 6.81·45-s + 0.867·47-s − 4.17·49-s − 13.3·51-s + 7.45·53-s − 21.3·57-s + 10.1·59-s − 12.7·61-s + ⋯ |
L(s) = 1 | − 1.80·3-s + 0.447·5-s + 0.635·7-s + 2.27·9-s + 0.152·13-s − 0.808·15-s + 1.03·17-s + 1.56·19-s − 1.14·21-s + 0.586·23-s + 0.200·25-s − 2.29·27-s + 1.90·29-s − 1.38·31-s + 0.284·35-s + 1.53·37-s − 0.274·39-s + 1.57·41-s + 0.906·43-s + 1.01·45-s + 0.126·47-s − 0.596·49-s − 1.87·51-s + 1.02·53-s − 2.82·57-s + 1.32·59-s − 1.62·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.780425818\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.780425818\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 3 | \( 1 + 3.13T + 3T^{2} \) |
| 7 | \( 1 - 1.68T + 7T^{2} \) |
| 13 | \( 1 - 0.548T + 13T^{2} \) |
| 17 | \( 1 - 4.26T + 17T^{2} \) |
| 19 | \( 1 - 6.81T + 19T^{2} \) |
| 23 | \( 1 - 2.81T + 23T^{2} \) |
| 29 | \( 1 - 10.2T + 29T^{2} \) |
| 31 | \( 1 + 7.71T + 31T^{2} \) |
| 37 | \( 1 - 9.36T + 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 - 5.94T + 43T^{2} \) |
| 47 | \( 1 - 0.867T + 47T^{2} \) |
| 53 | \( 1 - 7.45T + 53T^{2} \) |
| 59 | \( 1 - 10.1T + 59T^{2} \) |
| 61 | \( 1 + 12.7T + 61T^{2} \) |
| 67 | \( 1 - 7.39T + 67T^{2} \) |
| 71 | \( 1 + 4.54T + 71T^{2} \) |
| 73 | \( 1 - 13.6T + 73T^{2} \) |
| 79 | \( 1 + 0.638T + 79T^{2} \) |
| 83 | \( 1 + 2.81T + 83T^{2} \) |
| 89 | \( 1 + 10.1T + 89T^{2} \) |
| 97 | \( 1 + 5.07T + 97T^{2} \) |
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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
−7.42309542777364250623029642349, −6.99191201260988207070286466841, −5.94648874688592044497300604459, −5.80358175183819999470781982535, −5.01075685079944881241634593091, −4.58639747499799551792253264740, −3.58601393502385426814605649513, −2.48256024096529046252199053635, −1.16906813137937684730442949492, −0.903782929934779094805207480403,
0.903782929934779094805207480403, 1.16906813137937684730442949492, 2.48256024096529046252199053635, 3.58601393502385426814605649513, 4.58639747499799551792253264740, 5.01075685079944881241634593091, 5.80358175183819999470781982535, 5.94648874688592044497300604459, 6.99191201260988207070286466841, 7.42309542777364250623029642349