| L(s) = 1 | + 113·3-s + 1.15e3·5-s + 6.20e3·9-s − 1.46e4·11-s + 1.30e5·15-s + 5.31e5·23-s + 9.34e5·25-s − 3.98e4·27-s + 1.54e6·31-s − 1.65e6·33-s + 7.16e5·37-s + 7.14e6·45-s + 6.08e6·47-s + 5.76e6·49-s − 1.52e7·53-s − 1.68e7·55-s + 4.10e6·59-s − 1.98e7·67-s + 6.00e7·69-s − 7.04e6·71-s + 1.05e8·75-s − 4.52e7·81-s − 8.41e7·89-s + 1.74e8·93-s − 8.11e7·97-s − 9.08e7·99-s + 3.62e6·103-s + ⋯ |
| L(s) = 1 | + 1.39·3-s + 1.84·5-s + 0.946·9-s − 11-s + 2.56·15-s + 1.90·23-s + 2.39·25-s − 0.0750·27-s + 1.66·31-s − 1.39·33-s + 0.382·37-s + 1.74·45-s + 1.24·47-s + 49-s − 1.93·53-s − 1.84·55-s + 0.338·59-s − 0.982·67-s + 2.65·69-s − 0.277·71-s + 3.33·75-s − 1.05·81-s − 1.34·89-s + 2.32·93-s − 0.916·97-s − 0.946·99-s + 0.0322·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(5.250876136\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.250876136\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + p^{4} T \) |
| good | 3 | \( 1 - 113 T + p^{8} T^{2} \) |
| 5 | \( 1 - 1151 T + p^{8} T^{2} \) |
| 7 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 13 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 17 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 19 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 23 | \( 1 - 531793 T + p^{8} T^{2} \) |
| 29 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 31 | \( 1 - 1541233 T + p^{8} T^{2} \) |
| 37 | \( 1 - 716447 T + p^{8} T^{2} \) |
| 41 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 43 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 47 | \( 1 - 6080638 T + p^{8} T^{2} \) |
| 53 | \( 1 + 15265438 T + p^{8} T^{2} \) |
| 59 | \( 1 - 4101553 T + p^{8} T^{2} \) |
| 61 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 67 | \( 1 + 19806767 T + p^{8} T^{2} \) |
| 71 | \( 1 + 7043087 T + p^{8} T^{2} \) |
| 73 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 79 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 83 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 89 | \( 1 + 84100993 T + p^{8} T^{2} \) |
| 97 | \( 1 + 81155713 T + p^{8} T^{2} \) |
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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
−10.83806799625592627490607973741, −9.943936985043482990448084542304, −9.211640291206299761802025460826, −8.408358831887267882480427478682, −7.17155135592153231069536537169, −5.92179482464763470068082532987, −4.81652255258588167122481062919, −2.94784297030400541727535461960, −2.43759067344389067385810535106, −1.22367582453707044927984579175,
1.22367582453707044927984579175, 2.43759067344389067385810535106, 2.94784297030400541727535461960, 4.81652255258588167122481062919, 5.92179482464763470068082532987, 7.17155135592153231069536537169, 8.408358831887267882480427478682, 9.211640291206299761802025460826, 9.943936985043482990448084542304, 10.83806799625592627490607973741
