L(s) = 1 | − 4·5-s − 2·17-s + 2·19-s − 8·23-s + 11·25-s − 2·29-s − 4·31-s − 6·37-s − 2·41-s + 8·43-s − 4·47-s + 10·53-s + 6·59-s − 4·61-s − 12·67-s + 14·73-s − 8·79-s + 6·83-s + 8·85-s + 10·89-s − 8·95-s + 2·97-s + 12·101-s + 12·103-s + 12·107-s + 10·109-s − 6·113-s + ⋯ |
L(s) = 1 | − 1.78·5-s − 0.485·17-s + 0.458·19-s − 1.66·23-s + 11/5·25-s − 0.371·29-s − 0.718·31-s − 0.986·37-s − 0.312·41-s + 1.21·43-s − 0.583·47-s + 1.37·53-s + 0.781·59-s − 0.512·61-s − 1.46·67-s + 1.63·73-s − 0.900·79-s + 0.658·83-s + 0.867·85-s + 1.05·89-s − 0.820·95-s + 0.203·97-s + 1.19·101-s + 1.18·103-s + 1.16·107-s + 0.957·109-s − 0.564·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8274989252\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8274989252\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−8.515074793505794327617365568851, −7.69639740408856738387093436756, −7.36849069017085576948583776260, −6.46556088998900741811578342958, −5.49102837407369040609817160877, −4.54986168178647908951568994162, −3.89113776702465302721269609787, −3.28822043116364534811243543739, −2.03505353231784433825131603728, −0.51874703726271079374177335007,
0.51874703726271079374177335007, 2.03505353231784433825131603728, 3.28822043116364534811243543739, 3.89113776702465302721269609787, 4.54986168178647908951568994162, 5.49102837407369040609817160877, 6.46556088998900741811578342958, 7.36849069017085576948583776260, 7.69639740408856738387093436756, 8.515074793505794327617365568851