| L(s) = 1 | + 2.82·3-s + 5.00·9-s + 2.82·11-s − 6·17-s + 8.48·19-s + 5.65·27-s + 8.00·33-s + 6·41-s + 8.48·43-s − 7·49-s − 16.9·51-s + 24·57-s + 14.1·59-s − 8.48·67-s − 2·73-s + 1.00·81-s + 2.82·83-s + 18·89-s + 10·97-s + 14.1·99-s + 19.7·107-s − 18·113-s + ⋯ |
| L(s) = 1 | + 1.63·3-s + 1.66·9-s + 0.852·11-s − 1.45·17-s + 1.94·19-s + 1.08·27-s + 1.39·33-s + 0.937·41-s + 1.29·43-s − 49-s − 2.37·51-s + 3.17·57-s + 1.84·59-s − 1.03·67-s − 0.234·73-s + 0.111·81-s + 0.310·83-s + 1.90·89-s + 1.01·97-s + 1.42·99-s + 1.91·107-s − 1.69·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.225655767\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.225655767\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 2.82T + 3T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 - 2.82T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 - 8.48T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 8.48T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 14.1T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 8.48T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 2.82T + 83T^{2} \) |
| 89 | \( 1 - 18T + 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−8.004604059297186575675466799584, −7.47739101701819609266179145244, −6.86190371842249173263160904110, −6.02842313016901507347653475403, −4.96673371219222678923271082352, −4.15213607452696720572799923352, −3.54584521107838647336478443335, −2.77015956688253651923819653440, −2.03809384289219825909719242311, −1.04057642387582761234161837251,
1.04057642387582761234161837251, 2.03809384289219825909719242311, 2.77015956688253651923819653440, 3.54584521107838647336478443335, 4.15213607452696720572799923352, 4.96673371219222678923271082352, 6.02842313016901507347653475403, 6.86190371842249173263160904110, 7.47739101701819609266179145244, 8.004604059297186575675466799584
