| L(s) = 1 | + 1.67·3-s − 2.92·5-s − 1.65·7-s − 0.193·9-s + 5.49·11-s − 4.79·13-s − 4.90·15-s + 5.49·17-s − 0.255·19-s − 2.77·21-s + 3.87·23-s + 3.55·25-s − 5.34·27-s + 2.98·29-s − 5.08·31-s + 9.20·33-s + 4.85·35-s − 0.611·37-s − 8.03·39-s + 3.44·41-s − 3.26·43-s + 0.566·45-s − 2.23·47-s − 4.24·49-s + 9.20·51-s − 1.60·53-s − 16.0·55-s + ⋯ |
| L(s) = 1 | + 0.967·3-s − 1.30·5-s − 0.627·7-s − 0.0645·9-s + 1.65·11-s − 1.32·13-s − 1.26·15-s + 1.33·17-s − 0.0585·19-s − 0.606·21-s + 0.807·23-s + 0.711·25-s − 1.02·27-s + 0.554·29-s − 0.913·31-s + 1.60·33-s + 0.820·35-s − 0.100·37-s − 1.28·39-s + 0.538·41-s − 0.497·43-s + 0.0843·45-s − 0.325·47-s − 0.606·49-s + 1.28·51-s − 0.220·53-s − 2.16·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 - T \) |
| good | 3 | \( 1 - 1.67T + 3T^{2} \) |
| 5 | \( 1 + 2.92T + 5T^{2} \) |
| 7 | \( 1 + 1.65T + 7T^{2} \) |
| 11 | \( 1 - 5.49T + 11T^{2} \) |
| 13 | \( 1 + 4.79T + 13T^{2} \) |
| 17 | \( 1 - 5.49T + 17T^{2} \) |
| 19 | \( 1 + 0.255T + 19T^{2} \) |
| 23 | \( 1 - 3.87T + 23T^{2} \) |
| 29 | \( 1 - 2.98T + 29T^{2} \) |
| 31 | \( 1 + 5.08T + 31T^{2} \) |
| 37 | \( 1 + 0.611T + 37T^{2} \) |
| 41 | \( 1 - 3.44T + 41T^{2} \) |
| 43 | \( 1 + 3.26T + 43T^{2} \) |
| 47 | \( 1 + 2.23T + 47T^{2} \) |
| 53 | \( 1 + 1.60T + 53T^{2} \) |
| 59 | \( 1 + 8.61T + 59T^{2} \) |
| 61 | \( 1 + 2.48T + 61T^{2} \) |
| 67 | \( 1 + 8.51T + 67T^{2} \) |
| 71 | \( 1 + 15.2T + 71T^{2} \) |
| 73 | \( 1 + 0.465T + 73T^{2} \) |
| 79 | \( 1 + 0.500T + 79T^{2} \) |
| 83 | \( 1 - 8.22T + 83T^{2} \) |
| 89 | \( 1 + 0.0767T + 89T^{2} \) |
| 97 | \( 1 + 13.7T + 97T^{2} \) |
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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
−8.002880654552394337066268487621, −7.47632870151470462737599141054, −6.88791486375859140941413433916, −5.95415510813000264463156594770, −4.84941394628710301024714919173, −4.00552912956239074997441269459, −3.35901278753167101418096703538, −2.83878708161752543178362941167, −1.43661666931688568526009928458, 0,
1.43661666931688568526009928458, 2.83878708161752543178362941167, 3.35901278753167101418096703538, 4.00552912956239074997441269459, 4.84941394628710301024714919173, 5.95415510813000264463156594770, 6.88791486375859140941413433916, 7.47632870151470462737599141054, 8.002880654552394337066268487621
