| L(s) = 1 | − 3-s + 3·7-s − 2·9-s + 3·13-s + 3·17-s − 3·21-s − 9·23-s − 5·25-s + 5·27-s − 9·29-s + 6·31-s + 6·37-s − 3·39-s + 6·41-s − 8·43-s + 2·49-s − 3·51-s − 9·53-s + 3·59-s − 6·61-s − 6·63-s − 5·67-s + 9·69-s − 11·73-s + 5·75-s − 12·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.13·7-s − 2/3·9-s + 0.832·13-s + 0.727·17-s − 0.654·21-s − 1.87·23-s − 25-s + 0.962·27-s − 1.67·29-s + 1.07·31-s + 0.986·37-s − 0.480·39-s + 0.937·41-s − 1.21·43-s + 2/7·49-s − 0.420·51-s − 1.23·53-s + 0.390·59-s − 0.768·61-s − 0.755·63-s − 0.610·67-s + 1.08·69-s − 1.28·73-s + 0.577·75-s − 1.35·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.313534639\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.313534639\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−13.95732325243153, −13.34856799130245, −12.90650346169361, −12.09242243861493, −11.80564982184835, −11.39435899083117, −11.09355900741864, −10.41566779798014, −9.973303120123516, −9.406848719747610, −8.724603524163859, −8.251307597623409, −7.715519733567409, −7.598103134134901, −6.443370417470058, −6.075984748468175, −5.713465226762950, −5.166876362432107, −4.471961888563773, −4.014934460675873, −3.350578362632215, −2.599996746566386, −1.790020779095101, −1.379616303651164, −0.3794716953476557,
0.3794716953476557, 1.379616303651164, 1.790020779095101, 2.599996746566386, 3.350578362632215, 4.014934460675873, 4.471961888563773, 5.166876362432107, 5.713465226762950, 6.075984748468175, 6.443370417470058, 7.598103134134901, 7.715519733567409, 8.251307597623409, 8.724603524163859, 9.406848719747610, 9.973303120123516, 10.41566779798014, 11.09355900741864, 11.39435899083117, 11.80564982184835, 12.09242243861493, 12.90650346169361, 13.34856799130245, 13.95732325243153
