| L(s) = 1 | + 4·5-s − 2·7-s − 13-s + 3·17-s − 23-s + 11·25-s − 5·29-s − 4·31-s − 8·35-s − 2·41-s − 43-s + 12·47-s − 3·49-s − 9·53-s + 6·59-s − 5·61-s − 4·65-s + 6·67-s + 12·71-s − 6·73-s + 3·79-s + 4·83-s + 12·85-s + 2·91-s − 12·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 0.755·7-s − 0.277·13-s + 0.727·17-s − 0.208·23-s + 11/5·25-s − 0.928·29-s − 0.718·31-s − 1.35·35-s − 0.312·41-s − 0.152·43-s + 1.75·47-s − 3/7·49-s − 1.23·53-s + 0.781·59-s − 0.640·61-s − 0.496·65-s + 0.733·67-s + 1.42·71-s − 0.702·73-s + 0.337·79-s + 0.439·83-s + 1.30·85-s + 0.209·91-s − 1.21·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 226512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 226512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−13.26186472884007, −12.72238061491220, −12.43604191286832, −11.94272746619187, −11.06822560741868, −10.88422412058710, −10.18639326803160, −9.848539809664274, −9.559336570202599, −9.120533571074823, −8.656296168151827, −7.982875615121153, −7.345212577519393, −6.962639181527372, −6.336634235426341, −5.979714719708479, −5.569996290051190, −5.103919016829113, −4.555645422429341, −3.616686121102064, −3.367765740283387, −2.525131895019615, −2.199243732010420, −1.556450563232726, −0.9272887672005608, 0,
0.9272887672005608, 1.556450563232726, 2.199243732010420, 2.525131895019615, 3.367765740283387, 3.616686121102064, 4.555645422429341, 5.103919016829113, 5.569996290051190, 5.979714719708479, 6.336634235426341, 6.962639181527372, 7.345212577519393, 7.982875615121153, 8.656296168151827, 9.120533571074823, 9.559336570202599, 9.848539809664274, 10.18639326803160, 10.88422412058710, 11.06822560741868, 11.94272746619187, 12.43604191286832, 12.72238061491220, 13.26186472884007
