L(s) = 1 | + 2.62·2-s − 3-s + 4.87·4-s + 1.15·5-s − 2.62·6-s − 5.05·7-s + 7.53·8-s + 9-s + 3.03·10-s + 0.229·11-s − 4.87·12-s − 6.14·13-s − 13.2·14-s − 1.15·15-s + 9.99·16-s − 17-s + 2.62·18-s − 7.17·19-s + 5.65·20-s + 5.05·21-s + 0.602·22-s + 7.43·23-s − 7.53·24-s − 3.65·25-s − 16.1·26-s − 27-s − 24.6·28-s + ⋯ |
L(s) = 1 | + 1.85·2-s − 0.577·3-s + 2.43·4-s + 0.518·5-s − 1.07·6-s − 1.91·7-s + 2.66·8-s + 0.333·9-s + 0.961·10-s + 0.0692·11-s − 1.40·12-s − 1.70·13-s − 3.54·14-s − 0.299·15-s + 2.49·16-s − 0.242·17-s + 0.617·18-s − 1.64·19-s + 1.26·20-s + 1.10·21-s + 0.128·22-s + 1.54·23-s − 1.53·24-s − 0.731·25-s − 3.15·26-s − 0.192·27-s − 4.65·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 79 | \( 1 + T \) |
good | 2 | \( 1 - 2.62T + 2T^{2} \) |
| 5 | \( 1 - 1.15T + 5T^{2} \) |
| 7 | \( 1 + 5.05T + 7T^{2} \) |
| 11 | \( 1 - 0.229T + 11T^{2} \) |
| 13 | \( 1 + 6.14T + 13T^{2} \) |
| 19 | \( 1 + 7.17T + 19T^{2} \) |
| 23 | \( 1 - 7.43T + 23T^{2} \) |
| 29 | \( 1 + 3.53T + 29T^{2} \) |
| 31 | \( 1 - 0.658T + 31T^{2} \) |
| 37 | \( 1 - 0.679T + 37T^{2} \) |
| 41 | \( 1 + 0.445T + 41T^{2} \) |
| 43 | \( 1 + 4.01T + 43T^{2} \) |
| 47 | \( 1 + 0.773T + 47T^{2} \) |
| 53 | \( 1 + 9.98T + 53T^{2} \) |
| 59 | \( 1 + 9.96T + 59T^{2} \) |
| 61 | \( 1 - 4.77T + 61T^{2} \) |
| 67 | \( 1 - 4.61T + 67T^{2} \) |
| 71 | \( 1 + 13.2T + 71T^{2} \) |
| 73 | \( 1 - 3.80T + 73T^{2} \) |
| 83 | \( 1 + 14.6T + 83T^{2} \) |
| 89 | \( 1 - 5.86T + 89T^{2} \) |
| 97 | \( 1 - 5.66T + 97T^{2} \) |
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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
−7.44795547825063528290931458498, −6.92888190766777189117718468645, −6.31554384043687256468121921182, −5.91318682915458255541676073954, −5.01247027371169490336256268379, −4.42923009224438059015794833522, −3.47744909920100101368968851509, −2.75985463703684546622260906926, −2.00361549147656317744023318443, 0,
2.00361549147656317744023318443, 2.75985463703684546622260906926, 3.47744909920100101368968851509, 4.42923009224438059015794833522, 5.01247027371169490336256268379, 5.91318682915458255541676073954, 6.31554384043687256468121921182, 6.92888190766777189117718468645, 7.44795547825063528290931458498