| L(s) = 1 | + 4·2-s + 9·3-s + 16·4-s − 53.0·5-s + 36·6-s − 109.·7-s + 64·8-s + 81·9-s − 212.·10-s + 76.7·11-s + 144·12-s + 550.·13-s − 437.·14-s − 477.·15-s + 256·16-s − 1.73e3·17-s + 324·18-s + 2.70e3·19-s − 848.·20-s − 983.·21-s + 307.·22-s + 4.04e3·23-s + 576·24-s − 314.·25-s + 2.20e3·26-s + 729·27-s − 1.74e3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.948·5-s + 0.408·6-s − 0.842·7-s + 0.353·8-s + 0.333·9-s − 0.670·10-s + 0.191·11-s + 0.288·12-s + 0.903·13-s − 0.596·14-s − 0.547·15-s + 0.250·16-s − 1.45·17-s + 0.235·18-s + 1.71·19-s − 0.474·20-s − 0.486·21-s + 0.135·22-s + 1.59·23-s + 0.204·24-s − 0.100·25-s + 0.638·26-s + 0.192·27-s − 0.421·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.356715495\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.356715495\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4T \) |
| 3 | \( 1 - 9T \) |
| 59 | \( 1 - 3.48e3T \) |
| good | 5 | \( 1 + 53.0T + 3.12e3T^{2} \) |
| 7 | \( 1 + 109.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 76.7T + 1.61e5T^{2} \) |
| 13 | \( 1 - 550.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.73e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.70e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.04e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 7.12e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.67e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 4.94e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.33e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.19e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.44e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.53e4T + 4.18e8T^{2} \) |
| 61 | \( 1 + 1.18e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.83e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 7.88e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.00e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.55e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.36e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.44e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 5.81e4T + 8.58e9T^{2} \) |
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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
−10.99610669596170917399470078733, −9.575869723409545123639834461122, −8.817808026398242571589961325047, −7.60687419528407859729523406224, −6.91197724054992316673310375742, −5.76577659698756824728266279170, −4.35199026886272264430928390008, −3.58151803783987809156860947881, −2.65473352717323693755722326361, −0.888004521066059142337538096449,
0.888004521066059142337538096449, 2.65473352717323693755722326361, 3.58151803783987809156860947881, 4.35199026886272264430928390008, 5.76577659698756824728266279170, 6.91197724054992316673310375742, 7.60687419528407859729523406224, 8.817808026398242571589961325047, 9.575869723409545123639834461122, 10.99610669596170917399470078733
