L(s) = 1 | + i·4-s − i·9-s + 2·11-s
− 16-s + 2i·29-s + 36-s
+ 2i·44-s − i·64-s − 2·71-s
− 2i·79-s − 81-s − 2i·99-s
− 2i·109-s − 2·116-s + ⋯
|
L(s) = 1 | + i·4-s − i·9-s + 2·11-s
− 16-s + 2i·29-s + 36-s
+ 2i·44-s − i·64-s − 2·71-s
− 2i·79-s − 81-s − 2i·99-s
− 2i·109-s − 2·116-s + ⋯
|
\[\begin{aligned}
\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr
=\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s)
\end{aligned}
\]
\[\begin{aligned}
\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr
=\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s)
\end{aligned}
\]
\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]
where, for $p \notin \{5,\;7\}$,
\(F_p\) is a polynomial of degree 2. If $p \in \{5,\;7\}$, then $F_p$ is a polynomial of degree at most 1.
| $p$ | $F_p$ |
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - iT^{2} \) |
| 3 | \( 1 + iT^{2} \) |
| 11 | \( 1 - 2T + T^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 - 2iT - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 + 2T + T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 + 2iT - T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - iT^{2} \) |
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\[\begin{aligned}
L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}
\end{aligned}\]
Imaginary part of the first few zeros on the critical line
−9.756135287439598217163611926691, −8.927396946595284034690335471848, −8.706641390372644014055676846940, −7.36289440792070998940633942859, −6.79887453799926084924138946176, −6.05061216867643441238787335706, −4.60542854362445325517668098305, −3.76402609560450591347004108138, −3.13566014032118967832990293892, −1.48184834938861756288084290103,
1.29939866974928397743549768394, 2.34541662112690771483706212326, 3.95316090989407502307061453701, 4.68700314514062367333961334014, 5.76606802056393615800514373081, 6.40576646711482175871253270842, 7.26581164406950127297146803457, 8.330780849397573352821918570733, 9.202369470821391974638568171348, 9.808525302105704979753952440467