L(s) = 1 | − 2.54·2-s + 3-s + 4.45·4-s − 0.349·5-s − 2.54·6-s − 1.96·7-s − 6.24·8-s + 9-s + 0.887·10-s + 4.78·11-s + 4.45·12-s − 13-s + 5.00·14-s − 0.349·15-s + 6.95·16-s + 3.22·17-s − 2.54·18-s + 5.65·19-s − 1.55·20-s − 1.96·21-s − 12.1·22-s − 3.57·23-s − 6.24·24-s − 4.87·25-s + 2.54·26-s + 27-s − 8.77·28-s + ⋯ |
L(s) = 1 | − 1.79·2-s + 0.577·3-s + 2.22·4-s − 0.156·5-s − 1.03·6-s − 0.743·7-s − 2.20·8-s + 0.333·9-s + 0.280·10-s + 1.44·11-s + 1.28·12-s − 0.277·13-s + 1.33·14-s − 0.0901·15-s + 1.73·16-s + 0.781·17-s − 0.598·18-s + 1.29·19-s − 0.348·20-s − 0.429·21-s − 2.59·22-s − 0.746·23-s − 1.27·24-s − 0.975·25-s + 0.498·26-s + 0.192·27-s − 1.65·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
good | 2 | \( 1 + 2.54T + 2T^{2} \) |
| 5 | \( 1 + 0.349T + 5T^{2} \) |
| 7 | \( 1 + 1.96T + 7T^{2} \) |
| 11 | \( 1 - 4.78T + 11T^{2} \) |
| 17 | \( 1 - 3.22T + 17T^{2} \) |
| 19 | \( 1 - 5.65T + 19T^{2} \) |
| 23 | \( 1 + 3.57T + 23T^{2} \) |
| 29 | \( 1 + 6.87T + 29T^{2} \) |
| 31 | \( 1 + 3.27T + 31T^{2} \) |
| 37 | \( 1 + 9.23T + 37T^{2} \) |
| 41 | \( 1 + 6.19T + 41T^{2} \) |
| 43 | \( 1 + 8.42T + 43T^{2} \) |
| 47 | \( 1 - 3.74T + 47T^{2} \) |
| 53 | \( 1 + 2.63T + 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 + 14.8T + 61T^{2} \) |
| 67 | \( 1 - 6.74T + 67T^{2} \) |
| 71 | \( 1 - 5.34T + 71T^{2} \) |
| 73 | \( 1 - 0.574T + 73T^{2} \) |
| 79 | \( 1 - 9.42T + 79T^{2} \) |
| 83 | \( 1 - 3.35T + 83T^{2} \) |
| 89 | \( 1 - 0.450T + 89T^{2} \) |
| 97 | \( 1 + 6.47T + 97T^{2} \) |
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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
−8.149565912210791839020369767171, −7.55031491901715557745508014258, −6.94974494503628250594095796078, −6.31351443887431771893570334208, −5.33138287183788058246377920814, −3.71191718630469233817584103129, −3.34696855593149839249161971575, −2.03462109800072485414545279016, −1.33683372612908241998225450378, 0,
1.33683372612908241998225450378, 2.03462109800072485414545279016, 3.34696855593149839249161971575, 3.71191718630469233817584103129, 5.33138287183788058246377920814, 6.31351443887431771893570334208, 6.94974494503628250594095796078, 7.55031491901715557745508014258, 8.149565912210791839020369767171